Optical frequency comb generation in integrated lithium niobate devices

ABSTRACT

Kerr and electro-optic frequency comb generation in integrated lithium niobate devices is provided. In various embodiments, a microring resonator comprising lithium niobate is disposed on a thermal oxide substrate. The microring resonator has inner and outer edges. Electrodes are positioned along the inner and outer edges of the microring resonator. The electrodes are adapted to modulate the refractive index of the microring. A pump laser is optically coupled to the microring resonator. The microring resonator is adapted to emit an electro-optical frequency comb when receiving a pump mode from the pump laser and when the electrodes are driven at a frequency equal to a free-spectral-range of the microring resonator.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/664,806, filed on Apr. 30, 2018, which is hereby incorporated byreference in its entirety.

BACKGROUND

Embodiments of the present disclosure relate to optical frequency combgeneration, and more specifically, to Kerr and electro-optic frequencycomb generation in integrated lithium niobate devices.

BRIEF SUMMARY

According to embodiments of the present disclosure, methods and devicesfor optical frequency comb generation are provided.

In various embodiments, a device comprises: a thermal oxide substrate; amicroring resonator comprising lithium niobate, the microring resonatordisposed on the thermal oxide substrate; a pump laser optically coupledto the microring resonator, wherein the microring resonator is adaptedto emit a Kerr frequency comb when receiving a pump mode from the pumplaser.

In some embodiments, the Kerr frequency comb has a span of at least 80nm.

In some embodiments, the Kerr frequency comb has a span of at least 200nm.

In some embodiments, the Kerr frequency comb has a span of at least 300nm.

In some embodiments, the Kerr frequency comb has a span of at least 700nm.

In some embodiments, the microring resonator comprises a ridge portionextending from a slab portion, the ridge portion having a heightperpendicular to the slab portion and a width parallel to the slabportion.

In some embodiments, the slab portion has a thickness of 5 nm to 1000nm.

In some embodiments, the slab portion has a thickness of about 250 nm.

In some embodiments, the ridge portion has a height of 50 nm to 1000 nm.

In some embodiments, the ridge portion has a height of about 350 nm.

In some embodiments, the ridge portion has a width of 1000 nm to 5000nm.

In some embodiments, the ridge portion has a width of 1300 nm to 1500nm.

In some embodiments, the ridge portion has a cross sectional area of atmost 5 μm².

In some embodiments, the ridge portion has a cross sectional area of atmost 2 μm².

In some embodiments, the microring resonator has a pump rejection ratioof at most 47 dB.

In some embodiments, the pump mode has a wavelength of 300 nm to 5000nm.

In some embodiments, the pump mode has a wavelength of 1450 nm to 1600nm.

In some embodiments, the pump mode has a wavelength of 1500 nm to 1750nm.

In some embodiments, the pump mode has a wavelength of about 1570 nm.

In some embodiments, the microring resonator has a Q factor of at least500,000.

In some embodiments, the microring resonator has a Q factor of at least106.

In some embodiments, the microring resonator has a Q factor of at about107.

In some embodiments, the pump laser has a power of at least 50 mW.

In some embodiments, the pump laser has a power of at least 100 mW.

In some embodiments, the pump laser has a power of about 300 mW.

In some embodiments, the thermal oxide substrate comprises SiO₂.

In some embodiments, the microring resonator has a radius of 20 μm to2000 μm.

In some embodiments, the microring resonator has a radius of 20 μm to200 μm.

In some embodiments, the microring resonator has a radius of about 80μm.

In some embodiments, the Kerr frequency comb has a line spacing of about2 nm.

In some embodiments, the Kerr frequency comb has a TM-polarized comb.

In some embodiments, the Kerr frequency comb has a TE-polarized comb.

In various embodiments, a device comprises: a thermal oxide substrate; amicroring resonator comprising lithium niobate, the microring resonatordisposed on the thermal oxide substrate and having inner and outeredges; electrodes positioned along the inner and outer edges of themicroring resonator, adapted to modulate the refractive index of themicroring; a pump laser optically coupled to the microring resonator,wherein the microring resonator is adapted to emit an electro-opticalfrequency comb when receiving a pump mode from the pump laser and whenthe electrodes are driven at a frequency equal to a free-spectral-rangeof the microring resonator.

In some embodiments, the pump laser is optically coupled to themicroring resonator via a coupling ring resonator, the coupling ringresonator having a free spectral range that is a non-integer multiple ofa free spectral range of the microring resonator.

In some embodiments, an output waveguide is optically coupled to themicroring resonator.

In some embodiments, the coupling ring resonator has a free spectralrange greater than that of the microring resonator.

In some embodiments, the microring resonator is further adapted to emita Kerr frequency comb when receiving the pump mode from the pump laser.

In some embodiments, the electro-optical frequency comb spans at least10 nm.

In some embodiments, the electro-optical frequency comb has spacing of 1GHz to 300 GHz.

In some embodiments, the electro-optical frequency comb has spacing of10 GHz to 11 GHz.

In some embodiments, the electrodes comprise gold.

In some embodiments, the electrodes comprise copper.

In some embodiments, the microring resonator has a Q factor of at least500,000.

In some embodiments, the electrodes are positioned at least 1.5 μm fromthe edges of the microring resonator.

In some embodiments, the electrodes are positioned about 3.3 μm from theedges of the microring resonator.

In some embodiments, the thermal oxide substrate has a thickness ofabout 1 μm.

In some embodiments, the thermal oxide substrate has a thickness ofabout 2 μm.

In some embodiments, the electrodes are driven at a frequency of about10 GHz.

In some embodiments, the electrodes are driven at a power of about 10mW.

In some embodiments, the pump laser has a power of 0.1 mW to 3 W.

In some embodiments, the pump laser has a power of from 2 mW to 100 mW.

In some embodiments, the electro-optical frequency comb spans at least10 nm.

In some embodiments, the electro-optical frequency comb spans at least50 nm.

In some embodiments, the electro-optical frequency comb spans at least300 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 380 nm to 5000 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 1300 nm to 1700 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 1500 nm to 1680 nm.

In some embodiments, the microring resonator comprises a ridge portionextending from a slab portion, the ridge portion having a heightperpendicular to the slab portion and a width parallel to the slabportion.

In some embodiments, the slab portion has a thickness of 5 nm to 1000nm.

In some embodiments, the slab portion has a thickness of about 250 nm.

In some embodiments, the ridge portion has a height of about 350 nm.

In some embodiments, the ridge portion has a width of 100 nm to 5000 nm.

In some embodiments, the ridge portion has a width of 1300 nm to 2400nm.

In some embodiments, the ridge portion has a width of about 1400 nm.

In some embodiments, the ridge portion has a cross sectional area lessthan 5 μm².

In some embodiments, the ridge portion has a cross sectional area lessthan 2 μm².

In some embodiments, the microring resonator is air-clad.

In some embodiments, the microring resonator is clad with SiO₂.

In some embodiments, an inductor is electrically coupled to theelectrodes.

In some embodiments, the inductor is adapted to form a microwaveresonator having a resonant frequency, the resonant frequency being aninteger multiple of a free-spectral range of the microring resonator.

In various embodiments, a device comprises: a substrate; a microringresonator comprising an electro-optic material, the microring resonatordisposed on the substrate; a pump laser optically coupled to themicroring resonator, wherein the microring resonator is adapted to emita Kerr frequency comb when receiving a pump mode from the pump laser.

In some embodiments, the substrate comprises a thermal oxide.

In some embodiments, the substrate comprises SiO₂.

In some embodiments, the substrate comprises quartz.

In some embodiments, the substrate comprises sapphire.

In some embodiments, the electro-optic material comprises lithiumniobate.

In some embodiments, the electro-optic material comprises lithiumtantalate.

In some embodiments, the electro-optic material has an electro-opticcoefficient of at least 2 pm/V.

In some embodiments, the Kerr frequency comb has a span of at least 80nm.

In some embodiments, the Kerr frequency comb has a span of at least 200nm.

In some embodiments, the Kerr frequency comb has a span of at least 300nm.

In some embodiments, the Kerr frequency comb has a span of at least 700nm.

In some embodiments, the microring resonator comprises a ridge portionextending from a slab portion, the ridge portion having a heightperpendicular to the slab portion and a width parallel to the slabportion.

In some embodiments, the slab portion has a thickness of 5 nm to 1000nm.

In some embodiments, the slab portion has a thickness of about 250 nm.

In some embodiments, the ridge portion has a height of 50 nm to 1000 nm.

In some embodiments, the ridge portion has a height of about 350 nm.

In some embodiments, the ridge portion has a width of 1000 nm to 5000nm.

In some embodiments, the ridge portion has a width of 1300 nm to 1500nm.

In some embodiments, the ridge portion has a cross sectional area of atmost 5 μμm2.

In some embodiments, the ridge portion has a cross sectional area of atmost 2 μm2.

In some embodiments, the microring resonator has a pump rejection ratioof at most 47 dB.

In some embodiments, the pump mode has a wavelength of 300 nm to 5000nm.

In some embodiments, the pump mode has a wavelength of 1450 nm to 1600nm.

In some embodiments, the pump mode has a wavelength of 1500 nm to 1750nm.

In some embodiments, the pump mode has a wavelength of about 1570 nm.

In some embodiments, the microring resonator has a Q factor of at least500,000.

In some embodiments, the microring resonator has a Q factor of at least106.

In some embodiments, the microring resonator has a Q factor of at about107.

In some embodiments, the pump laser has a power of at least 50 mW.

In some embodiments, the pump laser has a power of at least 100 mW.

In some embodiments, the pump laser has a power of about 300 mW.

In some embodiments, the microring resonator has a radius of 20 μm to2000 μm.

In some embodiments, the microring resonator has a radius of 20 μm to200 μm.

In some embodiments, the microring resonator has a radius of about 80μm.

In some embodiments, the Kerr frequency comb has a line spacing of about2 nm.

In some embodiments, the Kerr frequency comb has a TM-polarized comb.

In some embodiments, the Kerr frequency comb has a TE-polarized comb.

In various embodiments, a device comprises: a substrate; a resonatorcomprising an electro-optic material, the resonator disposed on thesubstrate; electrodes positioned along the resonator with at least aportion of the resonator disposed between the electrodes, the electrodesadapted to modulate the refractive index of the resonator; a pump laseroptically coupled to the resonator, wherein the resonator is adapted toemit an electro-optical frequency comb when receiving a pump mode fromthe pump laser and when the electrodes are driven at a frequency, thefrequency being an integer multiple of a free-spectral-range of theresonator.

In some embodiments, the substrate comprises a thermal oxide.

In some embodiments, the substrate comprises SiO₂.

In some embodiments, the substrate comprises quartz.

In some embodiments, the substrate comprises sapphire.

In some embodiments, the electro-optic material comprises lithiumniobate.

In some embodiments, the electro-optic material comprises lithiumtantalate.

In some embodiments, the electro-optic material has an electro-opticcoefficient of at least 2 pm/V.

In some embodiments, the resonator comprises a racetrack resonator.

In some embodiments, the racetrack resonator has a minor axis measuring20 μm to 2000 μm and a perpendicular major axis measuring 0.1 mm to 20mm.

In some embodiments, the resonator has a minor axis measuring about 200μm and a perpendicular major axis measuring 2 mm to 6 mm.

In some embodiments, the major axis is perpendicular to an extraordinaryaxis of the electro-optic material.

In some embodiments, the resonator comprises a ring resonator.

In some embodiments, the resonator comprises a ring resonator or aracetrack resonator, the resonator has inner and outer edges, a firstsurface in contact with the substrate, and a second surface parallel tothe first surface, the electrodes are positioned along the first andsecond surfaces of the resonator.

In some embodiments, the resonator comprises a ring resonator or aracetrack resonator, the resonator has inner and outer edges, a firstsurface in contact with the substrate, and a second surface parallel tothe first surface, a first electrode is positioned along the outer edgeof the resonator, a second electrode is positioned along the secondsurface of the resonator.

In some embodiments, the pump laser is optically coupled to theresonator via a coupling resonator, the coupling resonator having a freespectral range that is a non-integer multiple of a free spectral rangeof the resonator.

In some embodiments, the coupling resonator comprises a ring resonator.

In some embodiments, an output waveguide is optically coupled to theresonator.

In some embodiments, the coupling resonator has a free spectral rangegreater than that of the resonator.

In some embodiments, the resonator is further adapted to emit a Kerrfrequency comb when receiving the pump mode from the pump laser.

In some embodiments, the electro-optical frequency comb spans at least10 nm.

In some embodiments, the electro-optical frequency comb has spacing of 1GHz to 300 GHz.

In some embodiments, the electro-optical frequency comb has spacing of10 GHz to 11 GHz.

In some embodiments, the electrodes comprise gold.

In some embodiments, the electrodes comprise copper.

In some embodiments, the resonator has a Q factor of at least 500,000.

In some embodiments, the electrodes are positioned at least 1.5 μm fromthe edges of the resonator.

In some embodiments, the electrodes are positioned about 3.3 μm from theedges of the resonator.

In some embodiments, the substrate has a thickness of about 1 μm.

In some embodiments, the substrate has a thickness of about 2 μm.

In some embodiments, the electrodes are driven at a frequency of about10 GHz.

In some embodiments, the electrodes are driven at a power of about 10mW.

In some embodiments, the pump laser has a power of 0.1 mW to 3 W.

In some embodiments, the pump laser has a power of from 2 mW to 100 mW.

In some embodiments, the electro-optical frequency comb spans at least10 nm.

In some embodiments, the electro-optical frequency comb spans at least50 nm.

In some embodiments, the electro-optical frequency comb spans at least300 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 380 nm to 5000 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 1300 nm to 1700 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 1500 nm to 1680 nm.

In some embodiments, the resonator comprises a ridge portion extendingfrom a slab portion, the ridge portion having a height perpendicular tothe slab portion and a width parallel to the slab portion.

In some embodiments, the slab portion has a thickness of 5 nm to 1000nm.

In some embodiments, the slab portion has a thickness of about 250 nm.

In some embodiments, the ridge portion has a height of about 350 nm.

In some embodiments, the ridge portion has a width of 100 nm to 5000 nm.

In some embodiments, the ridge portion has a width of 1300 nm to 2400nm.

In some embodiments, the ridge portion has a width of about 1400 nm.

In some embodiments, the ridge portion has a cross sectional area lessthan 5 μm².

In some embodiments, the ridge portion has a cross sectional area lessthan 2 μm².

In some embodiments, the resonator is air-clad.

In some embodiments, the resonator is clad with SiO₂.

In some embodiments, an inductor is electrically coupled to theelectrodes.

In some embodiments, the inductor is adapted to form a microwaveresonator having a resonant frequency, the resonant frequency being aninteger multiple of a free-spectral range of the microring resonator.

In various embodiments, a method of generating a Kerr frequency combcomprises: receiving a pump mode from a pump laser by a microringresonator, wherein the microring resonator comprising an electro-opticmaterial, the microring resonator disposed on a substrate, the pumplaser optically coupled to the microring resonator, the microringresonator is adapted to emit a Kerr frequency comb when receiving a pumpmode from the pump laser.

In some embodiments, the substrate comprises a thermal oxide.

In some embodiments, the substrate comprises SiO₂.

In some embodiments, the substrate comprises quartz.

In some embodiments, the substrate comprises sapphire.

In some embodiments, the electro-optic material comprises lithiumniobate.

In some embodiments, the electro-optic material comprises lithiumtantalate.

In some embodiments, the electro-optic material has an electro-opticcoefficient of at least 2 pm/V.

In some embodiments, the Kerr frequency comb has a span of at least 80nm.

In some embodiments, the Kerr frequency comb has a span of at least 200nm.

In some embodiments, the Kerr frequency comb has a span of at least 300nm.

In some embodiments, the Kerr frequency comb has a span of at least 700nm.

In some embodiments, the microring resonator comprises a ridge portionextending from a slab portion, the ridge portion having a heightperpendicular to the slab portion and a width parallel to the slabportion.

In some embodiments, the slab portion has a thickness of 5 nm to 1000nm.

In some embodiments, the slab portion has a thickness of about 250 nm.

In some embodiments, the ridge portion has a height of 50 nm to 1000 nm.

In some embodiments, the ridge portion has a height of about 350 nm.

In some embodiments, the ridge portion has a width of 1000 nm to 5000nm.

In some embodiments, the ridge portion has a width of 1300 nm to 1500nm.

In some embodiments, the ridge portion has a cross sectional area of atmost 5 μm².

In some embodiments, the ridge portion has a cross sectional area of atmost 2 μm².

In some embodiments, the microring resonator has a pump rejection ratioof at most 47 dB.

In some embodiments, the pump mode has a wavelength of 300 nm to 5000nm.

In some embodiments, the pump mode has a wavelength of 1450 nm to 1600nm.

In some embodiments, the pump mode has a wavelength of 1500 nm to 1750nm.

In some embodiments, the pump mode has a wavelength of about 1570 nm.

In some embodiments, the microring resonator has a Q factor of at least500,000.

In some embodiments, the microring resonator has a Q factor of at least10⁶.

In some embodiments, the microring resonator has a Q factor of at about10⁷.

In some embodiments, the pump laser has a power of at least 50 mW.

In some embodiments, the pump laser has a power of at least 100 mW.

In some embodiments, the pump laser has a power of about 300 mW.

In some embodiments, the microring resonator has a radius of 20 μm to2000 μm.

In some embodiments, the microring resonator has a radius of 20 μm to200 μm.

In some embodiments, the microring resonator has a radius of about 80μm.

In some embodiments, the Kerr frequency comb has a line spacing of about2 nm.

In some embodiments, the Kerr frequency comb has a TM-polarized comb.

In some embodiments, the Kerr frequency comb has a TE-polarized comb.

In various embodiments, a method of generating an electro-opticalfrequency comprises: receiving a pump mode from a pump laser by aresonator, wherein the resonator comprises an electro-optic material,the resonator is disposed on a substrate; driving electrodes at afrequency, the frequency being an integer multiple of afree-spectral-range of the resonator, wherein the electrodes arepositioned along the resonator with at least a portion of the resonatordisposed between the electrodes, the electrodes adapted to modulate therefractive index of the resonator.

In some embodiments, the substrate comprises a thermal oxide.

In some embodiments, the substrate comprises SiO₂.

In some embodiments, the substrate comprises quartz.

In some embodiments, the substrate comprises sapphire.

In some embodiments, the electro-optic material comprises lithiumniobate.

In some embodiments, the electro-optic material comprises lithiumtantalate.

In some embodiments, the electro-optic material has an electro-opticcoefficient of at least 2 pm/V.

In some embodiments, the resonator comprises a racetrack resonator.

In some embodiments, the racetrack resonator has a minor axis measuring20 μm to 2000 μm and a perpendicular major axis measuring 0.1 mm to 20mm.

In some embodiments, the resonator has a minor axis measuring about 200μm and a perpendicular major axis measuring 2 mm to 6 mm.

In some embodiments, the major axis is perpendicular to an extraordinaryaxis of the electro-optic material.

In some embodiments, the resonator comprises a ring resonator.

In some embodiments, the resonator comprises a ring resonator or aracetrack resonator, the resonator has inner and outer edges, a firstsurface in contact with the substrate, and a second surface parallel tothe first surface, the electrodes are positioned along the first andsecond surfaces of the resonator.

In some embodiments, the resonator comprises a ring resonator or aracetrack resonator, the resonator has inner and outer edges, a firstsurface in contact with the substrate, and a second surface parallel tothe first surface, a first electrode is positioned along the outer edgeof the resonator, a second electrode is positioned along the secondsurface of the resonator.

In some embodiments, the pump laser is optically coupled to theresonator via a coupling resonator, the coupling resonator having a freespectral range that is a non-integer multiple of a free spectral rangeof the resonator.

In some embodiments, the coupling resonator comprises a ring resonator.

In some embodiments, an output waveguide is optically coupled to theresonator.

In some embodiments, the coupling resonator has a free spectral rangegreater than that of the resonator.

In some embodiments, the resonator is further adapted to emit a Kerrfrequency comb when receiving the pump mode from the pump laser.

In some embodiments, the electro-optical frequency comb spans at least10 nm.

In some embodiments, the electro-optical frequency comb has spacing of 1GHz to 300 GHz.

In some embodiments, the electro-optical frequency comb has spacing of10 GHz to 11 GHz.

In some embodiments, the electrodes comprise gold.

In some embodiments, the electrodes comprise copper.

In some embodiments, the resonator has a Q factor of at least 500,000.

In some embodiments, the electrodes are positioned at least 1.5 μm fromthe edges of the resonator.

In some embodiments, the electrodes are positioned about 3.3 μm from theedges of the resonator.

In some embodiments, the substrate has a thickness of about 1 μm.

In some embodiments, the substrate has a thickness of about 2 μm.

In some embodiments, the electrodes are driven at a frequency of about10 GHz.

In some embodiments, the electrodes are driven at a power of about 10mW.

In some embodiments, the pump laser has a power of 0.1 mW to 3 W.

In some embodiments, the pump laser has a power of from 2 mW to 100 mW.

In some embodiments, the electro-optical frequency comb spans at least10 nm.

In some embodiments, the electro-optical frequency comb spans at least50 nm.

In some embodiments, the electro-optical frequency comb spans at least300 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 380 nm to 5000 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 1300 nm to 1700 nm.

In some embodiments, the electro-optical frequency comb has a centerwavelength of 1500 nm to 1680 nm.

In some embodiments, the resonator comprises a ridge portion extendingfrom a slab portion, the ridge portion having a height perpendicular tothe slab portion and a width parallel to the slab portion.

In some embodiments, the slab portion has a thickness of 5 nm to 1000nm.

In some embodiments, the slab portion has a thickness of about 250 nm.

In some embodiments, the ridge portion has a height of about 350 nm.

In some embodiments, the ridge portion has a width of 100 nm to 5000 nm.

In some embodiments, the ridge portion has a width of 1300 nm to 2400nm.

In some embodiments, the ridge portion has a width of about 1400 nm.

In some embodiments, the ridge portion has a cross sectional area lessthan 5 μm².

In some embodiments, the ridge portion has a cross sectional area lessthan 2 μm².

In some embodiments, the resonator is air-clad.

In some embodiments, the resonator is clad with SiO₂.

In some embodiments, an inductor is electrically coupled to theelectrodes.

In some embodiments, the inductor is adapted to form a microwaveresonator having a resonant frequency, the resonant frequency being aninteger multiple of a free-spectral range of the microring resonator.

In some embodiments, the ring resonator is driven in a traveling waveconfiguration.

In some embodiments, the ring resonator has a light propagationdirection, and the traveling wave configuration has a direction that isthe same as the light propagation direction.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

FIG. 1 is a cross sectional view of an exemplary waveguide according toembodiments of the present disclosure.

FIGS. 2A-C are plots of the group velocity dispersion (GVD) of anexemplary waveguide according to embodiments of the present disclosure.

FIG. 3A is a plot of simulated group velocity dispersion (GVD) of anexemplary integrated LN waveguide with various top widths (w) accordingto embodiments of the present disclosure.

FIG. 3B is a waveguide cross-section showing simulated field profile(E_(x)) of the fundamental TE mode according to embodiments of thepresent disclosure.

FIG. 3C is an image of an exemplary measurement setup according toembodiments of the present disclosure.

FIG. 3D is a schematic of an exemplary measurement setup according toembodiments of the present disclosure.

FIG. 3E is a plot showing the transmission spectrum and Lorentzian fitof the pump mode in an exemplary embodiment of the present disclosure.

FIG. 3F is a plot of the optical spectrum of a typical frequency combgenerated in the telecom wavelength range according to embodiments ofthe present disclosure.

FIG. 4A is a schematic of a microwave electrode geometry according toembodiments of the present disclosure.

FIG. 4B is a visualization of the sideband generation process accordingto embodiments of the present disclosure.

FIG. 4C is a microscope image of a fabricated lithium niobate resonatorwith microwave electrodes according to embodiments of the presentdisclosure.

FIG. 5 is an optical power spectrum of the EO comb, normalized to thefirst generated carrier according to embodiments of the presentdisclosure.

FIGS. 6A-B are schematic views of resonators configuration according toembodiments of the present disclosure.

FIG. 7 is a schematic view of a resonator configuration according toembodiments of the present disclosure.

FIG. 8 is a schematic view of a resonator configuration according toembodiments of the present disclosure.

FIG. 9 is a schematic view of electrode configurations according toembodiments of the present disclosure.

FIG. 10 is a cross-sectional view of electrode configurations accordingto embodiments of the present disclosure.

FIG. 11A is a cross-sectional view of electrode configurations accordingto embodiments of the present disclosure.

FIG. 11B is a schematic view of electrode configurations according toembodiments of the present disclosure.

FIG. 12 is a schematic view of an electrode configuration according toembodiments of the present disclosure.

FIG. 13 is a schematic view of resonator configurations according toembodiments of the present disclosure.

FIG. 14A is a schematic view of resonator configurations according toembodiments of the present disclosure.

FIG. 14B is a cross-sectional view of resonator configurations accordingto embodiments of the present disclosure.

FIG. 15 is a schematic view of a resonator configuration according toembodiments of the present disclosure.

FIG. 16 is a schematic view of a photonic circuit according toembodiments of the present disclosure.

FIG. 17 is a plot of the optical transmission spectrum of a microringresonator according to embodiments of the present disclosure.

FIG. 18 is a plot of the transmission spectra of a comb generatoraccording to embodiments of the present disclosure.

FIG. 19 is a plot of the numerically simulated group-velocitydispersions (GVD) for lithium niobate waveguides according toembodiments of the present disclosure.

FIG. 20A is a plot of the numerically simulated group-velocitydispersions (GVD) for lithium niobate waveguides according toembodiments of the present disclosure.

FIGS. 20B-C are plots of generated frequency comb spectra according toembodiments of the present disclosure.

FIG. 21A is a schematic view of an exemplary comb generator andmeasurement setup according to embodiments of the present disclosure.

FIGS. 21B-D are plots of the measured optical spectra at points along afilter according to embodiments of the present disclosure.

FIGS. 21E-F are eye diagrams of the output of a filter according toembodiments of the present disclosure.

FIG. 22 is a schematic view of an electro-optic comb generator accordingto embodiments of the present disclosure.

FIG. 23 is a visualization of a sideband generation process according toembodiments of the present disclosure.

FIG. 24 is a schematic view of an integrated microring electro-opticcomb generator according to embodiments of the present disclosure.

FIG. 25 is a micrograph of a fabricated lithium niobate microringresonator according to embodiments of the present disclosure.

FIG. 26 is a plot of output characteristics of a generated EO combaccording to embodiments of the present disclosure.

FIG. 27 is a plot of the measured electro-optic comb output spectrum andcalculated round-trip phase for various values of modulation frequencydetuning according to embodiments of the present disclosure.

FIG. 28 is a plot of the measured electro-optic comb output spectrum andcalculated round-trip phase for various values of optical frequencydetuning according to embodiments of the present disclosure.

FIG. 29 is a plot of the round trip phase for various electro-optic combgenerators according to embodiments of the present disclosure.

FIG. 30 is a plot of beats of an electro-optic comb and an experimentalsetup for their generation according to embodiments of the presentdisclosure.

FIGS. 31A-B are plots of the simulated dispersion and phase matchingcondition for an LN waveguide according to embodiments of the presentdisclosure.

FIGS. 32A-B are plots of the simulated dispersion and phase matchingcondition for a lithium niobate waveguide according to embodiments ofthe present disclosure.

FIGS. 33A-B are plots of the simulated dispersion and phase matchingcondition for a lithium niobate waveguide according to embodiments ofthe present disclosure.

FIG. 34 is a visualization of a sideband generation process according toembodiments of the present disclosure.

FIG. 35 is a visualization of a sideband generation process according toembodiments of the present disclosure.

FIG. 36 is a schematic view of a combined Kerr and electro-optic combgenerator according to embodiments of the present disclosure.

FIG. 37 is a plot of frequency combs generated by various combgenerators according to embodiments of the present disclosure.

FIG. 38 is a schematic view of a RE-EO comb generator based on a ringresonator according to embodiments of the present disclosure.

FIGS. 39A-B are plots of the intra-resonator power spectrum according toembodiments of the present disclosure.

FIGS. 40A-B are plots of the output spectra of a RE-EO comb generatoraccording to embodiments of the present disclosure.

FIG. 41 is a plots of the normalized linewidth correction factoraccording to embodiments of the present disclosure.

FIGS. 42A-B are plots of power spectra according to embodiments of thepresent disclosure.

FIG. 43 is a plot of time-averaged power transmission according toembodiments of the present disclosure.

FIGS. 44A-B are plots of output comb spectra according to embodiments ofthe present disclosure.

FIG. 45 is a plot of output comb spectra according to embodiments of thepresent disclosure.

FIG. 46 is a schematic view of a dual-ring EO comb generator accordingto embodiments of the present disclosure.

FIG. 47 is a plot of output spectra according to embodiments of thepresent disclosure.

FIG. 48 is a plot of output spectra according to embodiments of thepresent disclosure.

FIG. 49 is a schematic view of a WDM point-to-point inter-data centerlink according to embodiments of the present disclosure.

DETAILED DESCRIPTION

The present disclosure provides various devices and methods forgenerating frequency combs. Various embodiments include applications onlithium niobate (LN) and other χ(2) material based devices.

Integrated optical frequency combs are useful for precision timing,optical communication, and spectroscopy. Combs may be generated throughthe optical four-wave mixing process on chip where a powerful opticalpump laser is coupled to a high-Q optical resonator with high Kerr(χ(3)) nonlinearity. Alternatively, a frequency comb may be generatedelectro-optically (EO, χ(2)), where a coherent low-noise electricalsignal generates sidebands from the pump laser. In additional to LithiumNiobate, other χ(2) materials include lithium tantalate, PZT, andpotassium niobate.

Many frequency comb applications require, in addition to the combgenerator, a variety of photonic components such as fast switches,modulators, and/or nonlinear wavelength converters, which rely on strongsecond-order optical nonlinearity (χ⁽²⁾). These functionalities may beimplemented as discrete off-chip components, which come at the expenseof extra system complexity and increased losses. The present disclosureprovides for frequency comb generation on a single chip.

Type I Comb Generation: Kerr

The present disclosure provides Kerr frequency comb generation in highquality factor lithium niobate microresonators. In various embodiments,the generated combs span over 200 nm in the telecommunication wavelengthrange, and can be manipulated at high speed.

Kerr combs are generated on χ(3) materials such as silicon dioxide andsilicon nitride. The present disclosure describes structures thatgenerate Kerr combs on lithium niobate. For Kerr combs, ring resonatorsare designed to exhibit anomalous dispersion to maximize comb generationefficiency. The waveguide widths and heights are designed to achieveanomalous dispersion.

Microresonator Kerr frequency combs may be realized in various materialplatforms, including silica (SiO₂), silicon nitride (SiN), silicon (Si),crystalline fluorides, diamond, aluminium nitride (AlN), andaluminium-gallium arsenide (AlGaAs). While most of these materialspossess large χ⁽³⁾ nonlinearity and low optical loss, which are requiredfor Kerr comb generation, they usually have small or zero χ⁽²⁾nonlinearity and therefore are not suitable for on-chip integration ofχ⁽²⁾ components that are necessary for various frequency combapplications. Carrier-injection based Si devices can be electricallymodulated at high speeds, but exhibits much higher optical losses thantheir intrinsic Si counterparts. (Al)GaAs possesses high χ⁽²⁾nonlinearity for second harmonic generation, but much weakerelectro-optic effect (r₄₁=1.5×10⁻¹² m/V). As a result, off-chipcomponents are required for achieving these complex functionalities andon-chip manipulation of the generated combs is limited to slow thermaleffects or high-voltage electrical signals. Heterogeneous integration ofphotonic chips with different functionalities may be adopted tocircumvent this problem, however, this approach increases the complexityand cost of the system, and requires scalable and low-loss optical linksbetween chips.

The present disclosure provides for achieving χ⁽²⁾ functionalities bythe monolithic integration of lithium niobate nanophotonic waveguides,microring resonators, filters, and/or modulators on the same chip.Lithium niobate possesses large χ⁽²⁾ (r₃₃=3×10⁻¹¹ m/V) and χ⁽³⁾ (Kerr)(1.6×10⁻²¹ m²/V²) nonlinearities. The large χ⁽³⁾ nonlinearity enablesthe generation of a Kerr frequency comb, while the large χ⁽²⁾nonlinearity enables the manipulation of the generated comb by anexternally applied electric field.

In order for the χ⁽³⁾ optical parametric oscillation (OPO) process totake place, a microresonator with a high quality (Q) factor andanomalous dispersion is needed. The former ensures that the four-wavemixing (FWM) process could cascade and overcome the optical losses ofthe microresonator, and the latter compensates for the nonlinearresponses of the strong pump (self-phase modulation (SPM) andcross-phase modulation (XPM)). While ultra-high-Q (˜10⁸) LNwhispering-gallery-mode resonators may be fabricated using mechanicalpolishing methods, their dispersion properties are predetermined by thebulk material properties and cannot be engineered. In contrast, theintegrated approach of the present disclosure relies on an ultralow-lossmicro-structured LN photonic platform that offers dispersion engineeringcapability.

In various embodiments, a wide Kerr comb is generated on a LN photonicchip, spanning >700 nm, with electrically programmable filtering of asingle comb line with a pump rejection ratio of 47 dB, and intensitymodulation of a selected line at up to 500 Mbit s⁻¹.

Referring to FIG. 1, a cross section of an exemplary waveguide accordingto the present disclosure is provided. This exemplary waveguide is notclad, meaning that the waveguide is exposed to air at the top.

Referring to FIGS. 2A-C, the group velocity dispersion (GVD) of anexemplary waveguide without top cladding is plotted. FIG. 2A shows theGVD of an exemplary waveguide without top cladding for a transverseelectric (TE) mode. The GVD is plotted for various waveguide top widthsas a function of the input frequency, with each curve corresponding to adifferent waveguide width. The GVD is anomalous (negative) at certainfrequencies to achieve a Kerr comb. The GVD can also be engineered to benegative when there is a cladding (e.g., SiO₂). Waveguides withanomalous dispersion can then be fabricated into microring resonatorsfor Kerr comb generation. The principle allows for all opticalwavelengths supported by the material. FIG. 2B shows the minimum GVDobtained in FIG. 2A for each of the widths at which the GVD was plottedin FIG. 2A. FIG. 2C shows the GVD for each of the widths at which theGVD was plotted in FIG. 2A at the highest and lowest frequenciesmeasured in FIG. 2A.

A microresonator frequency comb is an excellent platform for broadbandcoherent light generation and precise frequency metrology. It provides acompact and inexpensive solution for a range of applications includingoptical clocks, pulse shaping, spectroscopy, and telecommunication.Using the Kerr χ(3) nonlinearity (four wave mixing, FWM), an on-chipfrequency comb can be generated in various material platforms and in awide wavelength range from visible to mid-infrared. However, thesematerials typically have zero (Si, SiN, SiO₂, chalcogenides) or small(AlN) electro-optic response. As a result, the generated combs can onlybe controlled at low frequency (thermal) or high voltage (40 V).

The present disclosure provides for Kerr frequency comb generation inintegrated lithium niobate (LN) microresonators that can be activelycontrolled at GHz frequency. LN on insulator platforms enable variouson-chip photonic devices including microresonators with quality (Q)factors up to 10′. In various embodiments, a dispersion engineeredmicro-ring resonator defined by dry etching is used, showing opticalfrequency comb generation in the telecom band spanning over 200 nm inwavelength.

Devices according to various embodiments comprise high-confinement LNmicroring resonators on top of thermal oxide substrate, fabricated usingelectron-beam lithography followed by an optimized dry etching process.The monolithic integration approach provides maximal freedom forwaveguide dispersion engineering.

FIG. 3A shows the simulated group velocity dispersion (GVD) ofintegrated LN waveguides with various top widths (w). FIG. 3B showssimulated field profile (E_(x)) of the fundamental TE mode for awaveguide with w=1300 nm. The waveguide 301 is made of LN, with a SiO₂substrate 302. FIG. 3C shows a camera image of an exemplary measurementsetup. The bright green spot 303 in FIG. 3C shows third harmonicgeneration (THG) light scattered from the device at high pump power.

FIG. 3D shows a schematic of an exemplary measurement setup. Lightpasses from tunable laser source 304 to erbium doped fiber amplifier 305to fiber polarization controller 306 to device under test 307 to opticalspectrum analyzer 308. TLS stands for tunable laser source. EDFA standsfor erbium doped fiber amplifier. FPC stands for fiber polarizationcontroller. DUT stands for device under test. OSA stands for opticalspectrum analyzer.

FIG. 3E shows transmission spectrum 309 (shown as rings) and Lorentzianfit 310 (shown as a curve) of the pump mode at ˜1570 nm, showing aloaded quality factor of 700,000. FIG. 3F shows the optical spectrum ofa typical frequency comb generated in the telecom wavelength range,spanning a wavelength band >200 nm.

FIG. 3A shows the calculated group velocity dispersion (GVD) for thefundamental TE mode in X-cut LN waveguides with a 600 nm thick devicelayer and various waveguide widths w. Anomalous dispersion, which isused for frequency comb generation, can be achieved in the telecomwavelength range (FIG. 3A). FIG. 3B shows the corresponding mode profile(E_(x) component) and device cross-section schematic for w=1300 nm,simulated using a Finite Difference Eigenmode (FDE) solver (Lumerical,Mode Solutions).

Exemplary devices are characterized using the setup shown in FIGS. 3C-D.In this example, pump light from a tunable telecom continuous-wave laser(Santec TSL-510) is amplified using an erbium doped fiber amplifier(EDFA, Amonics) and sent into the LN chip through a lensed fiber. Lightpolarization is controlled using an in-line fiber polarizationcontroller after the EDFA. Transmitted light at the output end of thechip is collected using a second lensed fiber and monitored with eitheran InGaAs photodetector (for transmission spectrum measurement) or anoptical spectrum analyzer (OSA, Yokogawa) (for comb spectrummeasurement).

FIG. 3E shows the transmission spectrum of the pump mode near 1570 nmmeasured at low power level, showing a loaded Q factor of 700,000. Thelowered Q factor is due to increased scattering loss in the air-claddeddevice. Nevertheless, this level of round-trip loss is sufficient tosupport the optical parametric oscillation (OPO) process at relativelylow pump power.

FIG. 3F shows the generated comb spectrum at an on-chip pump power of 50mW (17 dBm), with a repetition rate of ˜250 GHz. At this power level,the photorefractive induced instability inside the resonator is quenchedand the thermo-optic bistability dominates, making it possible to tunethe pump laser into the cavity resonance with fine control. Thefrequency comb spans over 200 nm in wavelength, and the envelope followsa hyperbolic secant function, indicating that the generated comb islikely in a soliton state. In this example, the repetition rate is toohigh for a direct time-domain measurement, which could be done in alarger resonator with a smaller free spectral range. The comb power bump311 at ˜1700 nm in FIG. 3F likely originates from the phase-matcheddispersive wave effect inside the resonator. Due to the high circulatingpower inside the ring resonator, second and third harmonic generation(SHG/THG) are observed simultaneously with the frequency combgeneration. The green spot 303 in FIG. 3C shows the scattered THG lightfrom the device in day light environment. The SHG light here isoverwhelmed by THG due to the lower camera sensitivity at ˜785 nm.Frequency combs at visible wavelengths are likely generated via theSHG/THG processes.

FIGS. 3A-F demonstrate frequency comb generation in a high Q (˜700,000)integrated LN microresonator. The high-confinement monolithic LNplatform allows engineering of waveguide dispersion and realizefrequency combs covering >200 nm wavelength range at a relatively lowpump power (˜50 mW). The frequency combs demonstrated here can becombined with the excellent electro-optic and nonlinear opticalproperties of LN, allowing for GHz active control of frequency combs andultra-wide spanning frequency comb generation (from visible to near IR).

In various embodiments, a single-crystal LN film of sub-micron thicknessis bonded on top of an SiO₂ substrate. By lithography and dry etching ofthe thin LN film, microresonators that have Q factors on the order of10⁶ can be realized. In various embodiments, an x-cut LN thin film waferis used to achieve anomalous dispersion in the telecom wavelength rangefor both the traverse electric (TE) and traverse magnetic (TM)polarizations. This can be achieved by carefully engineering thewaveguide width and thickness. In various embodiments, the dispersionengineered microresonator can have loaded and intrinsic Q factors of6.6×10⁵ and 1.1×10⁶, respectively, for the TE polarizations, with anestimated OPO pump threshold of ˜80 mW. In various embodiments,dispersion engineered microresonator can have loaded and intrinsic Qfactors of 6.0×10⁵ and 9.2×10⁵, respectively, for the TM polarizations.

In various embodiments, the microring resonator used in the Kerr combgenerator has a radius of 80 μm and a top width of 1.3 μm. In variousembodiments, a broadband frequency comb is generated for both TE-likeand TM-like polarization modes at a pump power of ˜300 mW in an inputbus waveguide, with a comb line spacing of ˜2 nm (250 GHz). In variousembodiments, the measured TM-polarized comb spectrum is ˜300 nm wide,while the TE-polarized comb spans from 1400 nm to 2100 nm.

In various embodiments, soliton states may be achieved by using temporalscanning techniques that have been deployed in other material platforms.

In various embodiments, the LN microresonators can sustain high opticalpowers (˜50 W of circulating power). Devices according to variousembodiments exhibit quenching behavior at high pump powers (>100 mW inthe waveguide), due to the photorefractive effect. This allows thethermal bistability effect to dominate, allowing stable positioning ofthe laser detuning with respect to cavity resonance. In variousembodiments, optical damage is not observed even after many hours ofoptical pumping, despite the high circulating power inside theresonators.

Referring to FIG. 16, a monolithic integrated photonic circuit accordingto embodiments of the present disclosure is shown. False-color scanningelectron microscope (SEM) image 1601 shows a fabricated lithium niobatenanophotonic circuit comprising a microresonator frequency combgenerator 1602 and an electro-optically tunable add-drop filter 1603, ata scale of 50 μm. Comb generator 1602 is air cladded to achieveanomalous dispersion, whereas the rest of the chip is cladded in SiO₂.Continuous wave pump light 1606 first passes through dispersionengineered microring resonator 1604 to generate a frequency comb. Thegenerated frequency comb is then filtered by add-drop microringresonator 1605. At the drop port of the filter, a single target combline is selected by applying an external bias voltage on the integratedelectrodes 1607 to align the filter passband with the comb line. Theselected comb line can be modulated at high speeds via the χ⁽²⁾ effect.The high χ⁽³⁾ nonlinearity of microresonator 1604 allows for thegeneration of Kerr frequency combs, while the high χ⁽²⁾ nonlinearity ofmicroresonator 1605 allows for the comb generated from comb generator1602 to be manipulated by an externally applied electric field.

Electrically tunable add-drop filter 1603 is integrated with combgenerator 1602 on the same chip 1601. In various embodiments, add-dropfilter 1603 comprises an LN microring resonator with a free spectralrange (FSR) designed to be ˜1% larger than comb generator 1602. Thisslightly detuned FSR utilizes the Vernier effect to allow for theselection of a single optical spectral line over a wide optical band.Filter ring 1605 is over-coupled to both the add and the drop buswaveguides with the same coupling strength, to ensure a high extinctionratio (on/off ratio). When the input light is on (off) resonance withthe filter, the majority of the optical power at the wavelength ofinterest will be transmitted to the drop (through) port of the filter.Microring filter 1605 is integrated with metal electrodes 1607positioned closely to the ring. This allows for fast and efficienttuning of the filter frequency, as well as amplitude modulation of thedropped light, via the electro-optic effect. In order to access themaximum electro-optic coefficient (r₃₃), the two resonators 1604 and1605 both operate in TE modes. Comb ring 1604 and filter ring 1605 arecladded with air and SiO₂, respectively, to ensure that both devicesoperate in their best configurations.

Numerical simulation shows that, for a device layer thickness of 600 nm,air cladding is necessary for anomalous dispersions. For the filterring, however, a SiO₂ cladding gives rise to a better electro-optictuning efficiency. Therefore, in various embodiments, the SiO₂ claddingin the comb generator area is removed, while the rest of the chip,including the filter ring, is cladded.

In various embodiments, devices are fabricated from a commercial x-cutLN-on-insulator (LNOI) wafer (NANOLN) with a 600-nm device layerthickness. Electron-beam lithography (EBL, 125 keV) is used to definethe patterns of optical waveguides and microring resonators in hydrogensilsesquioxane resist (FOX®-16 by Dow Corning) with a thickness of 600nm. The resist patterns are subsequently transferred to the LN filmusing Ar+-based reactive ion etching, with a bias power of ˜112 W, anetching rate of ˜30 nm min-1, and a selectivity of ˜1:1. The etchingdepth is 350 nm, with a 250-nm LN slab unetched. The coupling buswaveguide has a width of ˜800 nm, and the coupling gap is ˜800 nm. A1.5-μm-thick PMMA EBL resist is spun coated and exposed using a secondEBL with alignment, to produce the microelectrodes of the filter ringvia a lift-off process. The structures are then cladded with an800-nm-thick SiO₂ layer using plasma-enhanced chemical vapor deposition(PECVD). The oxide cladding in the comb generation areas is then removedthrough a photolithography step followed by hydrofluoric acid (HF) wetetching to realize air-cladded devices with the required anomalousdispersions. Finally, the chip edges are diced and polished to improvethe fiber-chip coupling.

Referring to FIG. 17, a plot of the optical transmission spectrum of amicroring resonator according to embodiments of the present disclosureis shown. FIG. 17 shows the transmission of the microring resonator 1904as a function of laser detuning, with loaded and intrinsic Q factors of6.6×10⁵ and 1.1×10⁶, respectively, for the TE polarization.

Referring to FIG. 18, a plot of the transmission spectra of a combgenerator according to embodiments of the present disclosure is shown.Top half 1808 and bottom half 1809 of the plot show the transmissionspectra at the through port with an applied DC bias voltage of 0 V and10 V, respectively, and a target comb line at 1616 nm. A zero bias, thecomb resonance 1811 has a 24-pm mismatch with the filter resonance 1810.Applying a bias voltage of 10 V can align the two resonances at 1812,with a measured electrical tuning efficiency of 2.4 pm V′.

Referring to FIGS. 19 and 20A, plots of the numerically simulatedgroup-velocity dispersions (GVD) for LN waveguides according toembodiments of the present disclosure are shown. The GVD is shown attelecom wavelengths for LN waveguides, with the different curvescorresponding to different top widths of the waveguides. Anomalousdispersion (GVD<0) can be achieved for both TM and TE modes, representedby FIG. 19 and FIG. 20A, respectively, based on the waveguide width andthickness. The waveguide dispersion diagrams and mode profiles in theseexamples are numerically calculated using a Finite Difference Eigenmode(FDE) solver.

Referring to FIGS. 20B-C, plots of the generated frequency comb spectraaccording to embodiments of the present disclosure are shown. FIG. 20Band FIG. 20C show the generated frequency comb spectra for TM and TEmodes, respectively, when the input laser is tuned into resonance withthe respective modes at a pump power of ˜300 mW in the bus waveguide.The generated combs have a line spacing of ˜250 GHz (˜2 nm), and span˜300 nm and ˜700 nm for the TM and TE modes, respectively. The envelopesof the comb spectra in FIGS. 20B-C indicate that the generated combs arenot in a soliton state, that is, are modulation instability frequencycombs.

Referring to FIG. 21A, a schematic view of an exemplary comb generatorand measurement setup according to embodiments of the present disclosureis shown. In the figure, AWG stands for arbitrary waveguide generator,EDFA stands for erbium-doped fiber amplifier, and PD stands forphotodetector. The selected target comb line at the drop port of filter2101 can be modulated at high speeds. An arbitrary waveform generator(AWG) is used to deliver random-binary voltage sequences to theelectrodes of the filter ring, in addition to the DC bias voltage.

Referring to FIGS. 21B-C, plots of the measured optical spectra atpoints along a filter according to embodiments of the present disclosureare shown. FIG. 21B and FIG. 21C correspond to the measured opticalspectra at the through and drop ports of filter 2101, respectively, witha picked out target comb line at 1616 nm. A DC bias voltage is appliedto align the filter frequency with a target comb line at 1616 nm. A pumpfrequency at ˜1556 nm exhibits a 730 pm mismatch with the filterresonance, resulting in an experimentally measured 47 dB rejection ofthe pump power in the drop port. The filter also exhibits a ˜20 dBextinction for the comb lines adjacent to the target line.

Referring to FIG. 21D, a plot of the measured optical spectra at pointsalong a filter according to embodiments of the present disclosure isshown. Different comb lines can be selected by applying different biasvoltages to filter 2101. FIG. 21D shows a zoomed in view of the dropport output spectra of the filter for DC bias voltages of 0 V (leftpeak) and 13 V (right peak). Applying a bias voltage of 13 V shifts thetarget from one comb line to the next one.

Referring to FIGS. 21E-F, eye diagrams of the output of the filteraccording to embodiments of the present disclosure are shown. Anarbitrary waveform generator (AWG) is used to deliver random-binaryvoltage sequences to the electrodes of the filter ring, and the realtime output optical power is monitored, resulting in the eye diagrams ofFIG. 21E and FIG. 21F, corresponding to data rates of 250 Mbit/s and 500Mbit/s, respectively. The peak to peak modulation voltage is 1.5 V,which is sufficient to tune the filter passband (˜3 pm wide) away fromthe target comb line. The scale bars in FIGS. 21E-F correspond to ascale of 1 ns. In various embodiments, the modulation speed can beimproved to beyond 100 Gbit/s by integrating a Mach-Zehnder modulatorafter the tuneable microring filter.

In various embodiments, frequency comb characterization is achieved witha continuous-wave (CW) light from a tuneable telecom laser (SantecTSL-510), amplified using an erbium-doped fiber amplifier (EDFA,Amonics). A 3-paddle fiber polarization controller is used to controlthe polarization of input light. Tapered lensed fibers are used tocouple light into and out from the waveguide facets of the LN chip. Theoutput light is sent into an optical spectrum analyzer (OSA, Yokogawa)for analysis. For filter control and manipulation, TE polarized modesare used to exploit the highest electro-optic tuning efficiency. DCsignals from a voltage supply (Keithley) and AC signals from anarbitrary waveform generator (AWG, Tektronix 70001A) are combined usinga bias T, before being sent to the filter electrodes using a high-speedground-signal (GS) probe (GGB Industries). The output optical signalfrom the drop port is sent to a 12-GHz photodetector (Newport 1544A),and analyzed using a 1-GHz real-time oscilloscope (Tektronix).

In various embodiments, electro-optic modulation can be embedded in thecomb generator, leading to active mode locking of a Kerr frequency comb.In various embodiments, the frequency comb source can be integrated witha multiplexer/demultiplexer and ultrafast electro-optic modulators onthe same chip to provide compact and low-cost dense-wavelength divisionmultiplexing (DWDM). This may be applied in ultra-broadband opticalfiber communication networks. Furthermore, fast and independent controlof the amplitude and phase of each comb line is useful for chip-scaleLiDAR systems, programmable pulse shaping and quantum informationprocessing.

Type II Comb Generation: Electro-Optic Comb

EO combs are generated based on χ(2) process, where light in a resonatoris phase modulated by an EO material. The modulation frequency closelymatches the free-spectral-range of the ring resonator. EO combs can begenerated using bulk crystal LN and off-chip cavities. The presentdisclosure provides for generating EO comb in on-chip lithium niobatewaveguide structures and photonic circuits based on that.

The migration of optical frequency comb generators to integrated devicesis motivated by a desire for efficient, compact, robust, and highrepetition-rate combs. Various approaches to on-chip frequency combgeneration rely on the Kerr (third-order, χ⁽³⁾) nonlinear opticalprocess, where a continuous wave (CW) laser source excites a low-lossoptical microresonator having a large Kerr nonlinear coefficient. Thisapproach enables wide-spanning Kerr frequency combs from the near- tomid-infrared in many material platforms such as silicon, silicondioxide, silicon nitride and magnesium fluoride. Sophisticated controlprotocols are typically required to keep Kerr combs stabilized.

An alternative frequency comb-generation method uses the electro-optic(EO) effect in materials with second-order (χ⁽²⁾) nonlinearity. EOfrequency comb generators can be created by passing a continuous wave(CW) laser through a sequence of discrete phase and amplitudemodulators. Such EO comb generators can feature high comb power and flatspectra, and can support flexible frequency spacing. They usually havenarrow frequency span, however, comprising only tens of lines andspanning only a few nanometers. Therefore, highly nonlinear fiber istypically required to further broaden the comb spectrum, increasing thesystem complexity and size. Broader EO combs can be generated using anoptical resonator to increase the nonlinear interaction strength.

Referring to FIG. 22, a schematic of an exemplary EO comb generatoraccording to embodiments of the present disclosure is shown. The EO combgenerator of FIG. 22 comprises an EO (χ²) phase modulator inside aFabry-Pérot (FP) resonator. In a such a resonator-based EO combgenerator, a CW laser is coupled to a bulk nonlinear crystal resonatorcontaining an EO phase modulator, and comb lines are generated solelythrough the χ² process.

A waveguide-based comb generator is shown in FIG. 24. A single-frequencyinput with electric field E_(in)(t)=Ê_(in) e^(iω) ⁰ ^(t) is coupled,with power coupling coefficient k and insertion loss γ, to a resonatorhaving round trip time T at center frequency ω₀ and round trip powerloss α. The resonator contains a phase modulator driven with modulationindex β and frequency ω_(m). The output electric field is

$\begin{matrix}{{E_{out}(t)} = {{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{E_{in}(t)}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\sum\limits_{n = 1}^{\infty}{r^{n}e^{{- i}\beta {F_{n}{({\omega_{m}t})}}}{E_{in}\left( {t - {nT}} \right)}}}}}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

where r=√{square root over ((1−γ)(1−k)α)} is the round trip electricfield transmission and F_(n)(ω_(m)t)=Σ_(i=1) ^(n) sin ω_(m)(t−iT) is themodulator coherence function.

The parameter l=1−r, corresponding to the round-trip electric fieldloss, is used in the main text for simplicity. When the optical carrieris resonant in the resonator (ω₀T=2πm₁) and the microwave drive signalis resonant (ω_(m)T=2πm₂), the modulator coherence function becomesF_(n)(ω_(m)t)=n sin ω_(m)(t−iT) and the output electric field can besimplified to

$\begin{matrix}{{E_{out}(t)} = {\left\lbrack {\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)} - {k\sqrt{\frac{1 - \gamma}{1 - k}}\frac{re^{{- i}{\beta \sin \omega}_{m}t}}{1 - {re^{{- {{i\beta}\sin \omega}_{m}}t}}}}} \right\rbrack {E_{in}(t)}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

This output electric field corresponds to an optical frequency combspaced at the modulation frequency. The power in the qth comb line awayfrom the center frequency can be found by rewriting Equation (3) as

$\begin{matrix}{{{E_{out}(t)} = {{{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{\overset{\hat{}}{E}}_{in}e^{i\omega_{0}t}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\sum\limits_{n = 1}^{\infty}{r^{n}e^{{- i}\; \beta \; {n\sin \omega}_{m}t}{\overset{\hat{}}{E}}_{in}e^{i{\omega_{0}{(t)}}}}}}} = {{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{\overset{\hat{}}{E}}_{in}e^{i\omega_{0}t}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\sum\limits_{q = {- \infty}}^{\infty}{{\overset{\hat{}}{E}}_{in}e^{{i{({\omega_{0} + {q\omega_{m}}})}}t}{\sum\limits_{n = 1}^{\infty}{r^{n}{J_{q}\left( {\beta \; n} \right)}}}}}}}}},} & {{Equation}\mspace{14mu} 3}\end{matrix}$

where J_(q) is the qth order Bessel function of the first kind. Thepower of the qth (nonzero) comb line is then

$\begin{matrix}{P_{q} = {k^{2}\frac{1 - \gamma}{1 - k}P_{in}{{\sum\limits_{n = 1}^{\infty}{r^{n}{J_{q}\left( {\beta \; n} \right)}}}}^{2}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

An approximation for the power of the qth comb as

$P_{q} \propto e^{- \frac{{q}{({1 - r})}}{\beta}}$

may be computed.

In the presence of optical and microwave detuning from resonance, thecomb spectrum can still be calculated. When the optical carrier is offresonance, the total round-trip phase is ω₀T=2πm₁+ϕ_(opt). Similarly,when the microwave carrier is off resonance the total round-trip phaseis ω_(m)T=2πm₂+ϕ_(micro). Using these expressions in Equation 1, we canfind the following expression for the power in the qth comb line:

$\begin{matrix}{P_{q} = {k^{2}\frac{1 - \gamma}{1 - k}P_{in}{{\sum\limits_{p = {- \infty}}^{\infty}{\overset{\infty}{\sum\limits_{n = 1}}{\left( {re^{i\; \varphi_{opt}}} \right)^{n}e^{ip\frac{\pi}{2}}{J_{q - p}\left( {\beta_{o}\left( {\varphi_{micro},n} \right)} \right)}{J_{p}\left( {\beta_{e}\left( {\varphi_{micro},n} \right)} \right)}}}}}^{2}}} & {{Equation}\mspace{14mu} 5}\end{matrix}$

The modified even and odd modulation indices (β_(e) and β_(o),respectively) are

$\begin{matrix}{{\beta_{e}\left( {\varphi_{micro},n} \right)} = {\beta\left\lbrack {{\frac{1}{2}\cot \varphi_{micro}\text{/}2} - \frac{{\cos \left( {n + \frac{1}{2}} \right)}\varphi_{micro}}{2\sin \varphi_{micro}\text{/}2}} \right\rbrack}} & {{Equation}\mspace{14mu} 6} \\{{\beta_{o}\left( {\varphi_{micro},n} \right)} = {\beta\left\lbrack {{- \frac{1}{2}} + \frac{{\sin \left( {n + \frac{1}{2}} \right)}\varphi_{micro}}{2\sin \varphi_{micro}\text{/}2}} \right\rbrack}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

In the regime of low optical detuning, the slope of the comb decreasesby a factor of cos(ϕ_(opt)). The effect of microwave detuning is harderto visualize, but results in a destructive interference condition forlarge values of q in Equation 5. This effect is demonstratedexperimentally and theoretically in FIG. 27.

The optical phase noise of the comb lines is important in applicationsthat require high optical signal-to-noise ratios, such as high-capacityoptical communications. It is well known that the optical phase noisecontribution from the pump laser does not increase with increasing combline index. By contrast, the phase noise contribution from the microwavemodulation signal increases in power with comb line quadratically withq. This can be shown by modifying the modulator coherence function toinclude the effects of microwave modulation phase noise θ(t):

$\begin{matrix}{{F_{n}\left( {\omega_{m}t} \right)} = {\sum\limits_{i = 1}^{n}{\sin {\omega_{m}\left( {t - {iT} + {\theta \left( {t - {iT}} \right)}} \right)}}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

The output optical field can then be written as:

$\begin{matrix}{{E_{out}(t)} = {{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{\overset{\hat{}}{E}}_{in}e^{i\omega_{0}t}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\overset{\hat{}}{E}}_{in}{\sum\limits_{q = {- \infty}}^{\infty}{\sum\limits_{n = 1}^{\infty}{r^{n}{J_{q}\left( {\beta n} \right)}{e^{{{i{({\omega_{0} + {q\omega_{m}}})}}t} + {iq{\theta {(t)}}}}.\left( {11} \right)}}}}}}} & {{Equation}\mspace{14mu} 9}\end{matrix}$

The phase noise amplitude increases linearly with increasing comb lineindex q, corresponding to a quadratic increase in phase noise power.

For applications that require few comb lines, this increase in microwavephase noise is often negligible because quartz crystal oscillators havevery low phase noise. For applications requiring many comb lines,however, the effect of microwave phase noise may be noticeable.Microwave phase noise suppression may be provided in EO comb generators.When the optical and microwave frequencies are resonant, higher ordercomb lines do not experience a quadratic increase in phase noise power.Instead, high frequency phase noise components are attenuated such thatthe high frequency phase noise is comparable for all comb lines.Furthermore, detuning the optical and microwave frequencies from theresonator FSR can further reduce the phase noise power. This indicatesthat EO comb generators can generate low-noise comb lines over theirentire dispersion-limited bandwidth. Additionally, integrated platforms,such as the one presented herein, enable additional filtering cavitiesand structures to be readily included in the resonator structure.

To include the effect of dispersion, a round-trip phase model isintroduced. In particular, the destructive interference that occurs dueto the microwave detuning motivates a phase-based resonanceapproximation for the viable comb bandwidth. A mathematical treatment ofthe dispersion limits of resonator-based EO comb generators in provided,including clarification of the physical interpretation of the round-tripphase model. Its application to combs of arbitrary bandwidth within agiven dispersion-limited window is demonstrated.

The resonance condition of an optical frequency ω_(q) in amicroresonator without EO modulation is |ω_(q)T−2πN|<2l, where the totalround-trip phase offset Δϕ_(q)=ω_(q)T−2πN, T=1/FSR is the round-triptime and N is the number of optical cycles per round-trip that ensuresthat |Δϕ_(q)|<2π. Frequency components outside of the resonance areattenuated by destructive interference, and thus do not resonate. Whenthe resonance condition is satisfied, the optical fields constructivelyinterfere inside the resonator at every time and spatial location.

In a resonator containing an EO phase modulator, the (nowtime-dependent) resonance condition becomes |Δϕ_(q)+β sin 2πf_(m)t|<21,where β is the modulation index and f_(m) is the modulation frequency.Here, it is clear that the resonance condition can be satisfied for muchlarger round-trip phase offsets Δϕ_(q) because within the round-tripresonator propagation time, the modulation term oscillates betweennegative and positive β (i.e. −β<β sin 2πf_(m)t<β).

This effect may be understood by plotting the total transmission of theEO comb generator for various β, as shown in 2603 of FIG. 26. Thetransmission is calculated by averaging the output power of atime-domain representation of the electric field given in Equation (3).The optical power output depends primarily on the interference betweenthe input optical field and the optical field inside the resonator. Asin a microresonator without EO modulation, the dips in the transmissionspectrum correspond to a large built-up field inside the resonator. Forvarious values of β, the width of the resonance increases, indicatingthat for large modulation indices, the resonance condition can besatisfied for various detuning values. As shown in FIG. 26, the amountof detuning is approximately equal to the modulation index β, as ispredicted by the phase model when Δϕ_(q)=ϕ_(opt).

The contributions to the optical phase offset Δϕ_(q) as a function offrequency can now be determined. The optical phase offset, as discussedpreviously, does not induce frequency-dependent phase shifts. However,microwave signal detuning and dispersion effects are frequencydependent.

Once the resonator has reached steady state, the output field is an EOcomb spaced at the modulation frequency f_(m), such that the qth combline frequency is f_(q)=f₀+qf_(m). A mismatch between the microwavefrequency and the resonator free spectral range, Δf_(m) results in afrequency-dependent phase offset ϕ_(micro)(q)=2πqΔf_(m)T.

For an arbitrary dispersion profile, it is possible to find thefrequency-dependent phase offset by integrating the group velocitydispersion profile of the waveguide. However, if the dispersion isapproximately linear with frequency, the dispersion-related phase offsetis Δϕ_(disp)(q)=2π(qf_(m))²β₂L where β₂L is the round-trip groupvelocity dispersion in fs²/mm.

A model for the total phase offset as a function of frequency to firstorder is obtained, Δϕ_(q)=Δϕ_(opt)+Δϕ_(micro)(q)+Δϕ_(disp)(q). In fact,this model agrees with alternative analytical models for the output combshape. In the case of maximum comb bandwidth, corresponding to zeromicrowave detuning and optical detuning satisfying ϕ_(opt)+β=0, themaximum dispersion-limited bandwidth is

${{\Delta f_{comb}} = {\frac{1}{\pi}\sqrt{\frac{2\beta}{\beta_{2}L}}}},$

agreeing with up to a factor of √{square root over (2)} due to thedifference in FSR of a Fabry-Pérot resonator and ring resonator ofidentical length.

Using this model, it is a straightforward optimization problem to startwith the frequency-dependent round-trip resonance condition and alterthe optical and microwave detuning so that the resonance condition issatisfied only for a desired frequency region, as is done to demonstratethe one-sided comb in FIG. 28.

Referring to FIG. 23, a visualization of the sideband generation processaccording to embodiments of the present disclosure is shown. A microwavesignal, with modulation frequency equal to the free spectral range ofthe optical resonator, couples light between different resonator modes.When the modulation frequency matches a harmonic of the resonator FSR,the optical sidebands generated by the phase modulator are resonant. Ina low-loss resonator, the light passes through the modulator many timesbefore being dissipated or coupled out, efficiently generating many comblines spaced at the modulation frequency. The modulation indexdetermines the strength of coupling between nearby frequency componentsafter passing through the modulator.

The output frequency comb can be predicted accurately by closed-formsolutions with spacings equal to the modulation frequency. The overallflatness of the comb strongly depends on the round-trip modulationstrength and the optical resonator loss. In particular, at frequenciesaway from the pump frequency, the comb line power decreasesexponentially: the optical power in the qth comb line is

${P_{q} \propto e^{\frac{|q|l}{\beta}}},$

where β=V_(p)/V_(π) is the phase modulation index, V_(p) is themicrowave drive peak amplitude, V_(π) is the half-wave voltage of thephase modulator,

$l = {\frac{\kappa}{FSR}\pi}$

is the round-trip electric-field loss coefficient of a resonator withdamping rate

${\kappa = \frac{\omega_{0}}{Q}},$

Q is the resonator quality factor, and ω₀ is the optical frequency.Strong phase modulation (large β) and a high-Q optical resonator (smalll) are therefore needed for generating flat and broad EO combs.Furthermore, dispersion sets a limit on the total comb bandwidth byintroducing frequency-dependent phase shifts that cause comb lines farfrom the pump frequency to fall out of resonance. EO frequency combsgenerated by free-space or fiber-based optical cavities are stilllimited to a few tens of nanometers of comb width by a combination ofweak modulation and limited dispersion engineering.

The present disclosure provides for overcoming these limitations bymonolithically integrating an EO comb generator on a thin film lithiumniobate (LN) nanophotonic platform. By leveraging the large χ⁽²⁾nonlinearity, strong microwave and optical field overlap, and ultra-lowloss optical waveguides enabled by this platform, EO combs withperformance superior to bulk EO comb generators are created. Compared toalternative integrated EO combs based on indium phosphide (InP) andsilicon (Si) platforms, where the effective EO modulation processes,created either by doping (Si) or operating near the material'sabsorption band edge (InP), induce high optical losses, embodiments ofthe present disclosure may achieve increases in comb width of nearly twoorders in magnitude.

Referring to FIG. 24, a schematic of an integrated microring EO combgenerator according to embodiments of the present disclosure is shown.The Fabry-Pérot resonator of FIG. 22 is replaced by a microringresonator that is EO-modulated at a frequency matching the FSR of themicroring. A continuous-wave laser coupled into the ring resonator isconverted to a frequency comb in the output optical waveguide.

In various embodiments, an EO frequency comb is generated with over 900unique frequencies spaced by 10.453 GHz, spanning 80 nm over part of thetelecommunication C-band, the entire L-band and part of the U-band.

Referring to FIG. 25, a micrograph of a fabricated LN microringresonator according to embodiments of the present disclosure is shown.The black lines 2501 are etched optical waveguides and the lighterregions 2502 are gold microelectrodes. The gold electrodes are driven sothat the phase shifts on the two sides of the microresonator areopposite, which is required to break the symmetry of different azimuthalorder optical modes, allowing for phase matching between the microwaveand circulating optical field, and enabling efficient frequencyconversion. A comb generator using this resonator uses a low-loss LNmicroring resonator with loaded Q of ˜1.5 million, which is integratedwith microwave electrodes for efficient phase modulation via the strongsecond-order nonlinearity of LN (r₃₃=30 pm/V). The tight confinement ofthe light (waveguide width=1.4 μm) allows for gold electrodes 2502 to beplaced only 3.3 μm away from the edge of the resonator, resulting inefficient microwave delivery to achieve strong phase modulation whilenot affecting the resonator Q factor.

In various embodiments, the EO comb generators are fabricated on x-cutsingle crystalline thin-film lithium niobate (LN) wafers (NANOLN). Thewafer stack comprises a 600 nm thin-film LN layer, a 2 μm thermallygrown SiO₂ layer and a 500 μm silicon handle layer. Standardelectron-beam (e-beam) lithography is used to pattern optical waveguideand micro-racetrack resonators. The patterns are then transferred intothe LN layer using argon (Ar⁺) plasma etching in an inductively coupledplasma reactive ion etching (ICP-RIE) tool. The etch depth is 350 nm,leaving a 250 nm thick LN slab behind, which enables efficient electricfield penetration into the waveguide core. Gold contact patterns arethen created using aligned e-beam lithography, and the metal istransferred using e-beam evaporation methods and lift-off processes. Thechip is then diced and the facets are polished for end-fire opticalcoupling. A 10 GHz FSR micro-racetrack measures 200 μm by 6.2 mm. Forillustration purposes, a 25 GHz FSR ring with otherwise the same designmeasuring 200 μm by 2.7 mm is displayed in FIG. 25, where the straightsection has a reduced length.

In various embodiments, a 10 GHz microwave drive signal is used. The 10GHz microwave drive signal is generated by a radio-frequency (RF)synthesizer and amplified by an electrical power amplifier. Theamplified electrical signal is passed through a microwave circulator anddelivered to the microelectrodes. As the microelectrodes represent acapacitive load, most of the electrical driving signal is reflected backto the circulator and terminated at the circulator output by a 50-Ωload.

Referring to FIG. 26, a plot of output characteristics of a generated EOcomb according to embodiments of the present disclosure is shown. An EOcomb generator is set up as described in FIG. 25, with an input opticalpower of 2 mW. The microresonator is modulated by an external microwavesynthesizer with peak voltage V_(p)=10 V (β=1.27π) at a frequency nearthe resonator FSR. The generated comb spectrum 2601 demonstrates abandwidth exceeding 80 nm and more than 900 comb lines with a slope of 1dB nm⁻¹. The signal-to-noise ratio of the comb lines exceeds 40 dB, butis limited by the noise floor and resolution of the optical spectrumanalyser. The left inset 2602 shows a magnified view of several comblines, with a line-to-line power variation of about 0.1 dB. The rightinset 2603 shows the measured transmission spectrum for severaldifferent modulation indices ft. When the modulation is turned on, theoptical resonance is broadened by twice the modulation index. Thisbehavior is predicted well by the round-trip phase model describedherein.

In various embodiments, light from a tunable laser (SANTEC TS510) islaunched into, and the comb output is collected from, the LN waveguidesby a pair of lensed optical fibers. The output comb is passed to anoptical spectrum analyser (OSA) having a minimum resolution of 20 pm.This finite resolution accounts for the limited signal-to-noise ratioobserved in FIG. 26 (˜20 dB). The shot-noise-limited signal-to-noiseratio is much higher, as the comb shot noise lies below the OSA noisefloor. Although the measurement in the present disclosure is chosen tocenter at 1600 nm, the frequency comb center wavelength can be flexiblychosen between 1500 nm to 1680 nm of the tunable laser's range withoutaffecting much of the generated comb width.

There are four resonator parameters that fully characterize the EO combspectrum: the internal round-trip transmission coefficient α, the powercoupling coefficient k, the coupler insertion loss of the coupler γ, andthe phase modulation index β. Finding each of these four parameters byfitting to the comb spectrum of the equation

$\begin{matrix}{P_{q} = {k^{2}\frac{1 - \gamma}{1 - k}P_{in}{{\sum\limits_{n = 1}^{\infty}{r^{n}{J_{q}\left( {\beta \; n} \right)}}}}^{2}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$

is difficult because the output comb can be fully determined by a subsetof these independent parameters (e.g., increasing the modulation indexhas the same effect as decreasing the loss in the resonator). Instead,each of the parameters must be measured separately. α and k may be foundby measuring the total transmitted power without phase modulation (see2603 in FIG. 26). Fitting to the expected transmission of an all-passring resonator results in values of Q=1.5×10⁶, α=0.95, and k=0.027. Thena grid search optimization for γ and β is performed, comparing themeasured output spectrum (FIG. 26) with the spectrum determined from theoutput time-domain electric field of equation

$\begin{matrix}{{E_{out}(t)} = {{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{E_{in}(t)}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\sum\limits_{n = 1}^{\infty}{r^{n}e^{{- i}\beta {F_{n}{({\omega_{m}t})}}}{E_{in}\left( {t - {nT}} \right)}}}}}} & {{Equation}\mspace{14mu} 11}\end{matrix}$

A best fit is found for γ=−0.004 dB and β=1.2 Tπ, where the averagedifference between experimental and theoretical comb line power is 0.6dB. The relative uncertainty in the measurement of β in this case is±4%, calculated by finding the furthest fit within a 95% confidenceinterval and calculating the resulting β. The output power transmissionfor nonzero modulation indices (2603 in FIG. 26) is calculated bysampling the output electric field with the above equation and averagingthe power over more than 100 modulation periods.

According to embodiments of the present disclosure, to achievewide-spanning EO combs, the waveguide dispersion is engineered such thatthe group velocity (or the FSR) of the ring is roughly a constant acrossthe entire frequency range. The dispersion of the waveguide wassimulated using finite element methods (LUMERICAL Mode Solutions). Thesimulation accounts for the LN material anisotropy and the finitewaveguide etching angle (around 70° from horizontal). The round-tripphase of the light inside the resonator is calculated by integrating thesimulated group velocity dispersion twice to determine the totalfrequency-dependent phase-shift. For various embodiments, with awaveguide ridge height of 350 nm, waveguide width of 1.4 μm, slabthickness of 250 nm, and SiO₂ top cladding of 1.5 μm, the dispersion ofthe waveguide is weakly normal and supports an EO comb cut-off bandwidthof ˜250 nm. It was found that for an air-cladded waveguide with a 600 nmthin-film LN layer, 550 nm etch depth and 1.8 μm waveguide width, a combspanning ˜1.3 octave can be generated with a round-trip modulationfrequency of 50 GHz and strength of β=1.2 π, as shown in FIG. 29. Thewaveguide dispersion can be tailored for low microwave drive powers atthe expense of a smaller comb span. For an air-cladded waveguide with a650 nm thin-film LN layer, etch depth of 620 nm and width 2400 nm, anoctave spanning comb can be generated with a phase modulation strengthof only 0.37τ. These results are presented in FIGS. 31-33

Referring to FIGS. 31A-B, plots of the simulated dispersion and phasematching condition for an LN waveguide according to embodiments of thepresent disclosure are shown. FIG. 31A shows the simulated dispersionfor an air-clad lithium niobate ridge waveguide with top width w=1,800nm, film thickness t=600 nm, and etch depth h=550 nm. FIG. 31B shows thephase matching condition for generating EO comb sidebands. Gray area3101 shows the region of phase matching, with round-trip modulationstrength β. With a 50 GHz microwave drive and =1.2 π, an EO combspanning 1.3 octaves can be generated.

Referring to FIGS. 32A-B, plots of the simulated dispersion and phasematching condition for an LN waveguide according to embodiments of thepresent disclosure are shown. FIG. 32A shows the simulated groupvelocity dispersion (GVD) for an air-clad lithium niobate waveguide witha different geometry optimized for an octave-spanning comb with smallmicrowave driving amplitude. The waveguide has top width w=2,400 nm,film thickness t=650 nm, and etch depth h=620 nm. FIG. 32B shows thephase matching condition for generating EO comb sidebands. Gray area3201 shows the region of phase matching, with round-trip modulationstrength β. With a 50 GHz microwave drive and =0.3 π, an octave spanningEO comb can be generated.

Referring to FIGS. 33A-B, plots of the simulated dispersion and phasematching condition for an LN waveguide according to embodiments of thepresent disclosure are shown. FIG. 33A shows the simulated dispersionfor an air-clad lithium niobate waveguide, dispersion engineered forbroad comb generation. The waveguide has top width w=2,700 nm, filmthickness t=660 nm, and etch depth h=630 nm. FIG. 33B shows the phasematching condition for generating EO comb sidebands. Gray area 3301shows the region of phase matching, with round-trip modulation strengthβ. With a 50 GHz microwave drive and =1.2 π, a broad EO comb spanningless than an octave can be generated in devices with small microwavemodulation amplitudes and high-Q optical resonators. Such EO combgeneration features a flat response over 600 nm.

In various embodiments, an EO comb generator features a directcapacitive drive electrode design, where the electrical powerconsumption P_(E) can be estimated as

P _(E)=½CV _(p) ²ω_(M)  Equation 12

where C≈200 fF is the estimated capacitance, V_(p) is the peak voltageand ω_(M) is the microwave frequency. For the broad comb shown in FIG.26, the calculated electrical power consumption is about 630 mW.

There are several ways to reduce the electrical power consumption of anEO comb generator according to embodiments of the present disclosure. Insome embodiments, the electrode gaps are not optimized and can bereduced to directly increase the electro-optic efficiency. A microwaveresonator with a quality factor of Q_(M) can be used to dramaticallyenhance the driving voltage, as only a narrow band microwave source isrequired. A microwave resonator has an enhanced voltage V_(p,eff) of

$\begin{matrix}{V_{p,{eff}} = \sqrt{\frac{2P_{E}Q_{M}}{\omega_{M}C}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

Comparing Equation 13 with Equation 12, the effective pumping power isincreased by a factor of Q_(M). This means that for a moderate Q_(M)=20at 10 GHz, the power consumption can be reduced to about 30 mW.

To estimate the minimum electrical power required to generate an octavespanning EO comb, a case in considered wherein the resonator is drivento 1.2V_(π) at 50 GHz FSR. Here, the capacitance of the device isreduced by a factor of 5 as the ring resonator becomes smaller toachieve a 50 GHz FSR. At the same time, the V_(π) also increases by afactor of 5 due to the shorter electrodes. For Q_(M)=20, the calculatedpower consumption is ˜750 mW. Through dispersion engineering and higheroptical Q microresonators, it is possible to achieve an octave spanningEO comb even at low drive voltages of V_(p)=0.3 V_(π). In this case, theelectrical power consumption is further reduced to only ˜45 mW.

A theoretical model is provided to quantify the fundamental limits ofthe wide spanning EO combs generated on an integrated platform. EO combspan in alternative approaches is limited to a narrow width by acombination of weak microwave modulation strength and native materialdispersion, which hinders the constructive interference needed forcascaded frequency conversion to generate comb lines far from the pumpfrequency. In contrast, the integrated EO comb generators of the presentdisclosure feature large modulation strength and the ability to engineerdispersion, which enables broader EO comb generation. To understand thelimitations of the EO comb generation process, the resonance conditionfor a comb line at optical frequency ω_(q) was analyzed. In atraditional resonator, the round-trip constructive interferencecondition is given by |Δϕ_(q)|<2l, where Δϕ=ω_(q)T−2πN is theaccumulated round-trip phase, T is the round-trip time, and N is thenumber of optical cycles per round-trip (chosen to minimize |Δϕ_(q)|).

For optical frequencies that satisfy this condition, the optical fieldinterferes constructively within the resonator. When the resonatorlength is modulated, as in an EO comb generator, the resonance conditionis modified into a dynamic one, where constructive interference occursperiodically at the microwave modulation frequency ω_(m) inside theresonator (i.e., |Δϕ_(q)+β sin ω_(m)t|<2l). Any frequency that does notsatisfy this dynamic resonance condition will halt the frequencyconversion process, thus limiting the comb width. This condition isreflected in the measured transmission spectrum of a microring resonatorunder microwave modulation (FIG. 26, right inset 2603). With nomicrowave modulation (β˜0), the transmission spectrum exhibits aLorentzian shape. By contrast, when the electrodes are stronglymodulated (large β) the half-width at half-maximum of the transmissionspectrum broadened by a factor of approximately β, confirming that thetolerable absolute accumulated phase |Δϕ_(q)| is increased to β. It istherefore clear that it is the strong phase modulation achieved in ourEO comb generator allowed for the continued cascade of phase modulationeven in the presence of dispersion.

To verify the round-trip phase model experimentally, the optical andmicrowave frequencies were detuned to generate different comb shapes andwidths. Referring to FIG. 27, a plot of the measured EO comb outputspectrum and calculated round-trip phase for various values ofmodulation frequency detuning according to embodiments of the presentdisclosure is shown. Plot 2710 shows three EO comb output spectra vswavelength for various values of modulation frequency detuning from theresonator free spectral range (Δω_(m)). The three plots, from top tobottom, correspond to a Δω_(m) of 10 MHz, 20 MHz, and 30 MHz,respectively.

Shaded region 2711 and envelope 2712 correspond to measured andnumerically simulated values, respectively. By increasing the microwavedetuning up to 30 MHz, a significant reduction in the comb frequencyspan was observed, which is predicted well by round-trip phase model2720. Calculated round trip phase model 2720 shows the round trip phaseΔϕ versus wavelength for the modulation frequency detuning values in2710. Gray shaded region 2721 highlights the constructive interferencecondition region beyond which EO comb generation is suppressed. Thisregion is bounded by ±β, the round trip modulation index.

Inset 2722 shows a zoomed out view of the round-trip phase versuswavelength plot 2720. The calculated cut-off frequency matches well withexperimental data, as shown by the dashed lines extending to 2710. Anyfrequency components having total accumulated phases larger than βcannot resonate, thus limiting the comb bandwidth.

Taking advantage of this dynamic resonance condition, asymmetric combscan be generated by appropriately choosing the optical and microwavedetuning. Referring to FIG. 28, a plot of the measured EO comb outputspectrum and calculated round-trip phase for various values of opticalfrequency detuning according to embodiments of the present disclosure isshown. Plot 2810 shows three EO comb output spectra versus wavelength inthe presence of both optical (Δω_(l)) and microwave (Δω_(m)) detuning.The three plots, from top to bottom, correspond to a Δω_(l) of 1 GHz, 2GHz, and 3 GHz, respectively, with all three having a Δω_(m) of 10 MHz.Shaded region 2811 and envelope 2812 correspond to measured andnumerically simulated values, respectively. By modifying the detuningvalues, different comb shapes, such as a single-sided EO comb, can begenerated. Calculated round trip phase model 2820 shows the round tripphase Δϕ versus wavelength for the modulation frequency detuning valuesin 2810. Gray shaded region 2821 highlights the constructiveinterference condition region beyond which EO comb generation issuppressed. This region is bounded by ±β, the round trip modulationindex. Inset 2822 shows a zoomed out view of the round-trip phase versuswavelength plot 2820. The calculated cut-off frequency matches well withexperimental data, as shown by the dashed lines extending to 2810. EOcombs driven off resonance, could be used as low-noise sources foroptical communications due to the noise-filtering properties of theoptical resonator.

Referring to FIG. 29, a plot of the round trip phase for various EO combgenerators according to embodiments of the present disclosure is shown.Plot 2900 shows simulated round-trip phase versus wavelength fortraditional bulk devices (curve 2901), a measured integrated device(curve 2902), and a dispersion engineered integrated device (curve2903). The simulations demonstrate that integrated EO combs can achievelarger dispersion-limited bandwidths than devices based on bulkcrystals, and dispersion engineering can enable octave spanning EOcombs. Traditionally, the span of EO comb generators is restricted bythe dispersion of bulk materials, whereas the EO comb generators of thepresent disclosure tightly confine light in optical waveguides, enablingfine tuning of dispersion. In various experiments, simulations haveshown that with a higher microwave modulation frequency of 50 GHz, ahigher optical pump power (currently only 2 mW in experiments conductedon embodiments of the present disclosure), and a dispersion engineeredLN rib waveguide resonator that minimizes variation in FSR, it ispossible to generate an EO comb spanning over an octave.

Referring to FIG. 30, a plot of beats of an EO comb and an experimentalsetup for their generation according to embodiments of the presentdisclosure are shown. An attractive property of EO comb generators aretheir excellent configurability and stability. Plot 3000 shows ameasured power spectral density on a logarithmic scale as a function offrequency. The measured power spectral densities of plot 3000 showcoherent beating of an EO comb according to embodiments of the presentdisclosure.

Experimental setup 3001 may be used to generate such results. Inexperimental setup 3001, the EO comb generator is driven by asuperposition of two phase-locked microwave signals with various valuesof frequency offset Δ. The comb generator is optically pumped close tozero detuning at a resonance near 1600 nm, and the optical output isdetected by a fast photodiode or other high-speed photodetector. Thebeat notes are detected by a radio-frequency spectrum analyzer. Thus,coherent beating between comb lines may be observed. Due to the strongphase modulation, this dual-driven EO comb contains frequency componentsfar beyond the ring resonator linewidth without modulation (120 MHz).Leveraging the high tolerance to the detuning of the modulationfrequency from the resonator FSR, the microresonator electrodes aredriven with two phase-locked microwave sources at various frequencyoffsets from 10.453 GHz, spanning over seven orders of magnitude,ranging from 10 Hz to over 100 MHz.

The ability to vary the frequency spacing of resonator-based EO combsover seven orders of magnitude is in stark contrast with Kerr-basedcombs, whose frequency offset is predetermined by the fabricatedresonator dimensions. Insets c, d, e, and f show magnified views of theindividual beat notes for various comb spacings on a linear frequencyscale. This demonstrates frequency components well beyond the resonatorbandwidth in the absence of modulation, confirms that phase modulationchanges the resonance condition to tolerate large microwave detuning.Additionally, this demonstrates the extreme flexibility in combfrequency spacing, which may enable applications requiringreconfigurable dynamic range, such as dual-comb spectroscopy orcomb-based ranging. In various embodiments, two independentmicroresonators can be integrated onto the same LN chip with highfabrication tolerance to avoid potential aliasing of the comb lines.

In the dual-drive EO comb generation experiment, two RF synthesizers arephase-locked via a common 10 MHz clock and are free to operate atdifferent frequencies. The two sinusoidal microwave signals are powerbalanced and combined using an RF power splitter and passed through theamplifier-circulator circuitry described previously. In the dual-driveEO comb measurements, the modulated light is passed to a fastphotodetector (New Focus 1544A) and the resulting electrical signal issent to a RF spectrum analyzer to record the beating in the RF domain.

Devices featuring high-Q microring resonators and highly confinedoptical waveguides for EO comb generation enable a new generation ofintegrated EO comb sources. Based on the demonstration of an EO combthat is almost two orders of magnitude larger than prior integrated EOcombs, dispersion engineering and high frequency modulation can enableefficient octave-spanning EO comb generators. The approachesdemonstrated herein can be used to realize EO combs all over the LNtransparency window, including visible and near-IR, simultaneously. Withthe added ability to integrate filters and resonators adjacent or insideEO comb generators on the same chip, comb line power, and hence SNR, canbe further increased by nearly 20 dB. Approaches set forth herein allowfor complex EO circuits to be integrated on the same chip, and thus areparticularly useful in microresonator frequency comb applications. Forexample, high-performance EO combs featuring high power and flat combsenable Tb/s optical communications links that rely on stable, low-noisecombs as sources for high capacity wavelength-division multiplexedsystems on a single chip. Furthermore, the EO comb generatordemonstrated herein provides many stable coherent optical frequencieswith electrically adjustable frequency spacing, paving the way forefficient dual-comb spectroscopy on a chip or highly-reconfigurablecomb-based ranging.

For the EO comb, ring resonators are designed to exhibit high qualityfactor. In various embodiments, a racetrack resonator is used tomaximize the section perpendicular to z-axis of the crystal. Theracetrack features an Euler curve connected circle to minimize opticalradiation loss when light propagates from the straight to the bentsections.

Electrodes are provided to achieve an anti-phase driving on the twooptical waveguides. The two electrodes inside the ring are connected,and the two outside are connected (This is opposite to a ring-modulatorconfiguration). In this configuration, a DC voltage has overall zeroeffect on the resonance frequency of the resonator as the top and thebottom waveguides have opposite phase shift. However, for a microwavesignal with a frequency that matches the circulation of light around theresonator, the microwave is phased-matched with light and can convertlight to adjacent optical modes. This process is cascaded for generatingthe EO comb.

In various embodiments, the microwave driving signal can be singlefrequency or multiple frequencies. The principle allows for all opticalwavelengths supported by the material. The electrodes can also bepositioned in a z-cut thin-film LN. In such cases, the electrode wouldstill have opposite connections for the two waveguides on two sides.Instead of an outside-inside relationship to the ring, the electrical +is above the ring, and the other side is below the ring.

In EO comb generation, the waveguide dispersion matters less, as themicrowave drive is strong, which overshadows the effect of dispersion.

Integrated optical frequency comb (OFC) generators are useful forapplications in precision timing, optical communication, andspectroscopy. One approach is nonlinear OFC generation, where a strongpump is coupled to a microcavity with high Kerr nonlinearity.

Another approach is to generate a frequency comb electro-optically (EO),which offers flexible input optical power and wavelength, low-noisecarrier generation and more stable operation in contrast to Kerr combgenerators. EO combs may be generated by coupling a single opticalcarrier into several phase and/or intensity modulators, such as in combgenerators based on asymmetric dual-driven Mach-Zehnder modulators.While these devices can be optimized to produce flat combs, the powerrequirement scales linearly with the number of desired carriers. Moreefficient EO comb generators based on resonant phase modulation may bedesigned, but are difficult to achieve on integrated platforms due tolarge dispersion, high optical loss and low EO efficiency.

The present disclosure provides for generating an EO comb spanning morethan 50 nm in the telecom L-band using an ultrahigh-Q lithium niobate(LiNbO₃) racetrack micro-resonator with integrated microwave electrodes.

FIG. 4A is a schematic of a microwave electrode geometry according toembodiments of the present disclosure. FIG. 4B is a visualization of thesideband generation process. High temporal confinement in high-Qresonators enhances cascaded sideband generation and flat comb output.FIG. 4C is a microscope image of a fabricated lithium niobate resonatorwith microwave electrodes.

FIGS. 4A-C illustrate the principle of operation of an exemplary deviceaccording to embodiments of the present disclosure. Optical microringresonator 401 is engineered to exhibit a periodic transmission spectrumwith nearly equal free-spectral-range (FSR), while microwave electrodes402 and 403 positioned along the circumference of the microringresonator modulate the refractive index of the ring (FIG. 4A). Invarious embodiments, the optical waveguide comprises lithium niobate orother electro-optic materials.

Electrodes 402 and 403 have opposite polarities, therefore, the indexshift achieved by the top half of the electrodes is opposite in sign tothe index shift achieved by the bottom half of the electrodes. Thisbreaks the orthogonality between different optical modes in the ringresonator, thereby facilitating conversion to other modes. Inalternative embodiments, mode conversion is achieved by only partiallymodulating the ring resonator, at the cost of halved efficiency.

Continuous-wave light is resonantly coupled into the cavity. When theelectrodes are driven at a frequency equal to the FSR of the cavity, thenewly generated optical carriers resonate in the cavity, enhancing thesideband generation process (FIG. 4B). For low-loss resonators, thepower in the m^(th) line away from the central carrier 404 isproportional to

${\exp \; \left( \frac{{- {m}}l}{\beta} \right)},$

where l is the round trip loss and β is the modulation index. Thisindicates that a higher drive voltage and a lower internal loss willresult in a flatter and broader comb.

A fabricated LiNbO₃ resonator is shown in FIG. 4C. It consists ofwaveguides 2 μm wide and 350 nm thick, residing on a 250 nm LiNbO₃ slab.Racetrack 405 comprises two straight arms that are 2.35 mm long,connected by two half-circles with a bending radius of 80 μm. The goldmicrowave electrodes 406 are placed 7 μm apart to preserve the opticalquality factor while delivering high EO efficiency. The device, afterintegrating the microwave electrodes, exhibits a loaded quality factorof 3.3 million (l=1.47%).

FIG. 5 is an optical power spectrum of the EO comb, normalized to thefirst generated carrier, showing excellent agreement with theoreticalprediction 501 (black dashed line). Calculated comb spectra with highermodulation indices β=0.5 π and 1.5 π (lines 502 and 503, respectively)show that an even flatter comb output is achievable.

FIG. 5 shows the output optical spectrum of the EO comb for a pumpwavelength of 1600 nm and ˜25 dBm microwave power at 23.93 GHz(modulation index β˜0.11π). The calculated envelope 501 of the powerspectrum is overlaid onto the experimental spectrum, showing excellentagreement. In addition, the theoretical output comb spectra at highermicrowave powers (larger β) are plotted (lines 502 and 503). This can beachieved by increasing the efficiency of the modulator, further reducingthe optical loss, and/or utilizing additional microwave amplifiers.

FIGS. 4A-C and FIG. 5 demonstrate an EO comb generator composed of anintegrated LiNbO₃ microring resonator with microwave electrodes. Thegenerated comb spectrum spans over 50 nm and matches well with thetheoretical model. With improved design and higher driving voltage, itis possible to generate even wider EO combs. This integrated EO combplatform enables frequency comb generation at flexible wavelengths andwith predictable comb line powers, opening paths forward for noveloptical communication schemes and spectroscopy applications.

An integrated EO comb uses a relatively strong microwave driving power(>1 W). To mitigate the power requirements, the following design can beimplemented (on-chip).

In FIG. 6, traveling microwave 601 is used as the driving electrodes.The travelling wave electrodes are designed in the same way as atraveling wave modulator, where the microwave co-propagates with lightaround resonator 602.

The microwave signal travels together with the light around themicroring resonator with similar group velocities. At crossover point603, the electrodes cross over each other, ensuring that the indexinduced on top and bottom halves of the ring resonator have the sameelectric field direction, thereby inducing refractive index changes ofthe same sign in both halves. This is in contrast to the non-travelingwave design 604 and that shown in FIG. 4A.

In this configuration, a much higher microwave bandwidth can beachieved, circumventing the RC bandwidth limit of the existing design.

In FIG. 7, LC microwave resonators are used for efficiency microwavedriving. Another design involves using a LC microwave resonator forenhanced microwave driving. Since only a narrowband microwave source isused, a high Q microwave resonator can amplify the microwave drivingstrength. This design can also be combined with the travelling wavedesign to simultaneously achieve strong and high frequency microwavemodulation.

Combining χ(2) and χ(3) Combs.

EO comb and Kerr comb maybe achieved on the same device. In this device,the EO comb is fabricated as described above, and the waveguides arealso designed to have anomalous GVD. The comb may be operated witheither a strong microwave driving strength (EO comb dominated), orstrong optical power (Kerr comb dominated) or anywhere in between (adynamic combination or synchronization of any stable output states)where both EO and Kerr effects are strong.

The combined χ(2) and χ(3) comb has several benefits. In a Kerr comb, itis difficult to achieve locking (phase coherence between each lines). AnEO comb can address this. An EO comb is difficult to achieve over alarge bandwidth. A Kerr comb can address this. Synchronizing the twocombs can allow for low noise, low power operations, and provide amethod to synchronize remote frequency combs through a common microwavesource.

χ(2) and χ(3) Combs Integrated Circuits

The integration of EO and Kerr combs on lithium niobate (or other χ(2))material allows for full scale active photonic circuits to beconstructed. In comb generation materials other than LN, fast modulationcannot be achieved. Only LN integrated circuits can achieve both EO,Kerr and fast modulation at the same time. This enables comb generationand fast modulation on the same chip. Comb generation and nonlinearconversion on the same chip may include periodically-poled lithiumniobate (PPLN) waveguides.

Referring to FIG. 8, integration of array of filters and modulators forcoherent telecommunication applications is illustrated. Here the combsare generated on the first device 801 (EO, Kerr or EO+Kerr). Then thecomb is passed through a series of filters 802 (here is ring add-dropfilters) where individual comb lines are picked out. These lines may beindividually processed (e.g., here modulated) through integrated LNelectro-optic modulators 803 and then used or recombined usingwavelength-division multiplexing for high bandwidth data communication.

Using a similar geometry, filters and modulators may be used for quantumentanglement control and generation.

The integrated combs may have soliton optical outputs (light bullets),that can be used for ranging and sensing. Here the comb would beintegrated with arrays of phase modulators on the same chip and anoutput device (e.g., grating) for ranging.

Comb generating structure may be integrated with PPLN structures toallow converting comb frequencies on chip or in-situ.

Referring to FIG. 34, a visualization of a sideband generation processaccording to embodiments of the present disclosure is shown. Thissideband generation process is characteristic of Kerr combs, which relyon a spontaneous four-wave mixing process that is efficient to generatebroad combs. In this process, two pump photons are converted to a signaland idler photon through the spontaneous process. Yet, the spontaneousprocess necessities a threshold to generate combs and the combs respondhighly nonlinearly with respect to input optical power.

Referring to FIG. 35, a visualization of a sideband generation processaccording to embodiments of the present disclosure is shown. Thissideband generation process is characteristic of EO combs, which rely ona seeded sum and difference frequency generation mechanism, whichindicates a threshold-less comb generation process. Frequency combs aregenerated from the electro-optic effect, where the new frequencies aregenerated by sum and difference frequency generation between one opticalpump photon and one microwave pump photon. However, to generate a broadEO comb, very strong EO modulation is required as the cascading EOprocess is inherently inefficient for generating a broad comb comparedto the Kerr process.

Kerr combs based on χ⁽³⁾ nonlinearity and electro-optic frequency combsbased on χ⁽²⁾ nonlinearity can be achieved separately on integratedlithium niobate (LN) nanophotonic platform, owing to LN's low loss andsimultaneous strong material χ⁽²⁾ and χ⁽³⁾ nonlinearities. Amicroresonator with both EO and Kerr nonlinearities could combine thebenefits of each mechanism and generate broad combs at low microwave andoptical powers.

In the present disclosure, frequency combs generated from a microringresonator with simultaneous Kerr and electro-optic nonlinearities aredisclosed. Experimental results show that in a tested resonator, thecombined Kerr and EO processes can produce a broader comb than eachindividual process alone.

Referring to FIG. 36, is a schematic of a combined Kerr and EO combgenerator according to embodiments of the present disclosure. In variousembodiments, a micro-racetrack resonator is fabricated on a x-cut thinfilm lithium niobate on insulator platform. In this embodiment, theresonator is designed to combine dispersion engineering andelectro-optic microwave drives to enable simultaneous Kerr and EOnonlinearity. The micro-racetrack resonator is designed to have afree-spectral-range (FSR) of ˜30 GHz, which is accessible by microwavedrivers. Typical loaded quality factors of such micro-racetracks areabout 2×10⁶.

Referring to FIG. 37, a plot of frequency combs generated by variouscomb generators according to embodiments of the present disclosure isshown. In various experiments, a microring resonator was investigatedunder solely EO drive, at a low optical pump power of ˜2 mW. Noobservable optical nonlinearity was induced. With the same resonator, amicrowave drive at 10 mW and an optical drive of 2 mW, frequency comb3701 is generated. The frequency comb is a narrow and single sided EOcomb spanning 9 nm with sharp cutoff frequencies. This is due to thestrong negative dispersion of the waveguide and mode crossing near thepump wavelength. Frequency comb 3702 is the result of driving by only astrong optical pump. The same microring resonator is used with a highoptical pump power of ˜100 mW, with no EO drive, and the outputted comb3702 contains frequencies generated by only Raman shift and Kerrprocesses. A strong Raman peak is seen, shifted by ˜55 nm to the redside of the pump at 1570 nm. The Raman peak further generated amini-comb spanning ˜8 nm through the Kerr process. When the EO drive isset to 10 mW and the optical pump power is set to ˜100 mW, combs 3703and 3704, spanning over 300 nm, observed. Due to the high amplifiedspontaneous emission (ASE) noise in the EDFA used in the experiment, thecomb lines are measured above an optical noise floor of ˜−60 dBm. Combs3703 and 3704 have the same microwave and optical driving powers, butare measured on different optical spectrum analyzers. Thus, EO and Kerrpumped resonators have the potential to generate broader frequency combsthat that beyond the span of either mechanism alone.

Referring to FIG. 9, various embodiments of an electrode configurationfor an integrated lithium niobate microring resonator are shown. Blacklines 901 indicate optical waveguides. Light shaded region 902 and darkshaded region 903 indicate metal electrodes, with the light shadedregion and the dark shaded region having opposite electrical polarity.One electrode of a certain polarity is placed on the inside of thewaveguide structure 901, and another electrode of an opposite polarityis placed on the outside of the waveguide structure, allowing for anelectric field to be present between them. The electric field formedbetween these two electrodes will have a significant component along thecrystal's extraordinary axis, indicated by the direction of arrow 904.If the component of the electric field parallel to the extraordinaryaxis is in the same direction as the axis, then it will induce apositive index change, and if the component is antiparallel to the axis,then it will induce a negative index change. Thus, in the embodiments ofFIG. 9, the electrodes are configured such that the induced index in thetop half of optical waveguide is opposite in sign to the induced indexin the bottom half. Configurations 905 and 906 contain electrodes onlyon a portion of the ring resonator, while configurations 907 and 908have electrodes on both the top and bottom portions of the resonator.

Referring to FIG. 10, cross sectional views of two electrodeconfigurations are shown. In this diagram, the crystal's extraordinaryaxis, represented by the direction of arrow 1001 is perpendicular to thewafer plane. Lightly shaded region 1004 and darkly shaded region 1005represent electrodes, the lightly shaded region having a polarityopposite that of the darkly shaded region. One electrode of a certainpolarity is placed underneath the ring resonator 1006, and anotherelectrode of an opposite polarity is placed above the ring resonator,allowing for an electric field to be present between them with acomponent either parallel or antiparallel to crystal's extraordinaryaxis 1001. In embodiment 1002, electrodes are placed above and below thering resonator on two portions of the ring resonator, such that theelectric field generated in the two portions are in opposite directionto each other. In alternative embodiment 1003, electrodes are onlyplaced above and below the ring resonator on one portion of the ringresonator, allowing for partial modulation.

Referring to FIG. 11A, cross sectional views of two electrodeconfigurations are shown. In this diagram, the crystal's extraordinaryaxis, represented by the direction of arrow 1101 is perpendicular to thewafer plane. Lightly shaded region 1104 and darkly shaded region 1105represent electrodes, the lightly shaded region having a polarityopposite that of the darkly shaded region. One electrode of a certainpolarity is placed above the ring resonator 1106, and another electrodeof an opposite polarity is placed to the side of the ring resonator,allowing for an electric field to be present between them with acomponent either parallel or antiparallel to crystal's extraordinaryaxis 1101. This embodiment of the electrode configuration allows foreasier fabrication. In embodiment 1102, electrodes are placed above andto the side of the ring resonator on two portions of the ring resonator,such that the electric field generated in the two portions are inopposite direction to each other. In alternative embodiment 1103,electrodes are only placed above and to the side of the ring resonatoron one portion of the ring resonator, allowing for partial modulation.

Referring to FIG. 11B, top views of the embodiments of FIG. 11A areshown. In this diagram, the crystal's extraordinary axis, represented bythe direction of arrow 1107 is perpendicular to the wafer plane, and isshown as coming out of the plane of the page. Electrode configurations1108 and 1109 correspond to the partially modulated electrodeconfiguration 1103 in FIG. 11A, while electrode configurations 1110 and1111 correspond to the electrode configuration 112 of FIG. 11A.

Referring to FIG. 12, a schematic of an electrode configuration isshown. A microwave inductor is used in conjunction with a capacitordriver to achieve microwave resonances that improve the microwavedriving efficiency. The capacitor driver is placed around waveguide1201, and interfaces with external electrical circuits for input andoutput via a channel represented by dotted lines 1202.

Referring to FIG. 13, a schematic of a traveling wave configuration isshown. In this diagram, the crystal's extraordinary axis is indicated byarrow 1301. Lightly shaded regions 1304 and darkly shaded region 1305represent electrical transmission lines, with the lightly and darklyshaded regions wired to ground and an input signal, respectively, viaelectrical connections 1307. One of the ground transmission lines 1304is placed inside waveguide ring 1306, and the other is placed outside ofit. Signal transmission line 1305 crosses over the waveguide ring inbending region 1308, so that a portion of it is inside the waveguidering and a portion is outside of it. This ensures that the index shiftinduced in the top and bottom regions of the waveguide ring have thesame sign. In configuration 1302, the electrical transmission lines onlypartially cover the waveguide ring. In configuration 1303, theelectrical transmission lines are configured in a closed loop, and theexternal connections 1309 are connected to an inductive or capacitivecoupling element that ensures that the microwave driving signalpropagates in a direction matching that of the optical circulatingsignal in the waveguide.

Referring to FIGS. 14A-B, a schematic view of a traveling waveconfiguration is shown. In this diagram, the crystal has a z-cutconfiguration, with its extraordinary axis indicated by arrows 1401 and1408. Lightly shaded regions 1404 and darkly shaded region 1405represent electrical transmission lines, with the lightly and darklyshaded regions wired to an input signal and ground, respectively, viaelectrical connections 1406. The electrical transmission lines areplaced with the signal transmission line 1404 either above, below, or tothe side of the waveguide ring 1407, and the ground transmission lineplaced such that the electric field between the two transmission lineshas a component parallel to the crystal's extraordinary axis. Thiscreates an index shift that is in the same direction throughout affectedportions of the waveguide ring. Referring to FIG. 14A, configurations1402 and 1403 vary in the number of transmission lines used and thelength of the waveguide covered by the transmission lines. Referring toFIG. 14B, configurations 1409 and 1410 correspond to configurations 1402and 1403, with configuration 1410 having a transmission line placed tothe side of the waveguide, as opposed to below, which allows for easierfabrication. In configurations 1411 and 1412, the transmission linesonly partially cover the circumference of the waveguide ring, theconfiguration 1412 being easier to fabricate.

Referring to FIG. 15, a schematic of a combined EO/Kerr comb is shown.The ring 1501 is dispersion engineered to have Kerr nonlinearity, andelectrodes 1502 are also placed around the waveguide to achieve the EOeffect.

Optical frequency combs are useful for many different areas of science,ranging from sensing to telecommunications. One desirable comb featureis high optical power, which often reduces the effects of noise. Alimitation on comb generators based on microresonators, such as lithiumniobate ring resonators, is their efficiency in generating the comb. Inparticular, in order to create flat and broad combs, the interactionbetween light in the input waveguide and light inside of the resonatormust be small. However, this effect results in less light in the outputcomb.

A way to address this limitation is to try to efficiently trap lightinside the microresonator, where the comb is formed. Ideally, this wouldinvolve a coupler that was totally transparent at the input lightwavelength, but very reflective at all other wavelengths. This way, theinput laser is efficiently coupled into the resonator, while the newlygenerated comb inside the resonator cannot escape, due to the highreflectivity at the other wavelengths. However, a coupler with theseproperties could be extremely difficult to make and is not a practicalsolution.

A solution provided below is to use a ring as a wavelength-dependentcoupler. By tuning the circumference of the small ring, there is acircumference for which the input laser is efficiently coupled into thelarger ring, while other wavelengths are reflected. The details of thiseffect are discussed below. This design requires an additional waveguidefor the output comb. This is provided in this design because the combcannot travel back through the small ring coupler.

While a ring coupler is extremely useful, and easy to fabricate, thereare design trade-offs. For example, this design increases the power insome parts of the comb by 14 dB, a significant improvement. However, forcomb wavelengths far away from the input, the power can be the same oreven smaller. This effect occurs due to the fact that there are limitson how small of a ring coupler can be made. All rings have a propertycalled the free spectral range (FSR), which in this context is a measureof the total wavelength span over which the ring is periodic. In otherwords, any ring will be transparent to multiple wavelengths, the FSR isa measure of how far apart those wavelengths are. For the designdescribed below, the FSR limits the total width of the comb because onceother wavelengths can travel through the ring coupler, the efficiency ofthe comb generation process decreases. While a finite FSR does limit thepower in some parts of the comb, this effect can also be used to ouradvantage. By designing or tuning the FSR of the small ring to be anon-integer multiple of the large ring FSR, the comb can be broadened.

Optical frequency combs have uses ranging from metrology and precisiontime-keeping to spectroscopy and optical communications. Often, thesevaried applications require combs with vastly different characteristics.For example, precision timing applications require combs that span afull octave, while applications in spectroscopy often require combswhose frequency spacing can be easily change. For use in opticalcommunications, combs may have narrower width than in other applicationsbut must be flat and have high optical power.

Optical frequency combs can be generated by several different methods.Mode-locked lasers, for example, can output wide combs in differentwavelength ranges. Frequency combs can also be generated throughparametric generation, via the χ(3) nonlinearity in optical fibers orresonant structures. Comb generators based on high-Q nonlinearresonators have desirable output properties. However, the formationdynamics of these comb generators is complex and their noise propertiesare still not fully understood. Finally, flat and high-power combsuseful for optical communications can be generated by electro-optic (EO)modulation of a single-frequency optical field, but the powerconsumption of these comb generators is often too high.

Resonator-enhanced electro-optic (RE-EO) comb generators, which couplelight into a free-space or fiber-based resonators containing an EOmodulator, have been studied for over four decades and are moreefficient than comb generators based on cascaded modulation. Early RE-EOcomb generators, implemented in lossy free-space resonators with bulkycomponents are sensitive to fluctuations in the input optical frequencyand modulation frequency, increasing the locking requirements of thecomb generator. Low-loss integrated technologies enable RE-EO combgenerators whose modulation frequency can equal the resonator freespectral range (FSR), corresponding to a different regime of operation.The effects due to a non-resonant input optical frequency and modulationfrequency have been discussed in experimental contexts, but an exactanalytical form for the output has not been determined.

Additionally, low coupling between the input optical field and theresonator is crucial to ensure that the intra-resonator optical field ismodulated many times before being output-coupled, but results inconversion efficiencies less than 5%. Nevertheless, free-space RE-EOcomb generators with higher conversion efficiencies have beenexperimentally demonstrated by including an additional couplingresonator before the comb-generating resonator. While this concept iscommon for free-space comb generators, a dual-resonator design tailoredto integrated ring-resonators is not well-known.

The present disclosure provides for analysis of the output spectrum andnoise properties of a ring-based RE-EO comb generator for resonant andnon-resonant operation, i.e., when the optical and modulationfrequencies are resonant and non-resonant with the FSR, respectively. Tomodel frequency-dependent propagation such as dispersion, two numericalmodels to determine the output comb spectrum were developed andvalidated. To increase the output optical power of the comb, a dual-ringRE-EO comb generator is proposed that is composed of a small couplingring, which traps light at the input optical frequency, and a largercomb-generating ring that contains a phase modulator. According to someembodiments, this comb generator design offers an average increase incomb line power of 14 dB and meets the optical signal-to-noise ratio(OSNR) requirements of an inter-data center wavelengthdivision-multiplexed (WDM) optical communications link.

Referring now to FIG. 38, a RE-EO comb generator based on a ringresonator is shown. Light is coupled into the resonator with couplerpower transmission k and power insertion loss γ. In the resonator withFSR ω_(r) the light experiences round trip power loss (1−α) and ideal,lumped phase modulation with modulation frequency ω_(m) and modulationindex β. The intra-resonator field E_(c)(t) is discussed below. Asingle-frequency optical field E_(in)(t)=Ê_(in)e^(iω) ⁰ ^(t) is coupledinto a resonator that contains a phase modulator. Once inside thelow-loss resonator, light passes through the phase modulator many times,accumulating a sinusoidal time dependent phase, before beingoutput-coupled into the original waveguide. The complex output opticalfield, E_(out)(t), can be expressed as an infinite sum of time-shifted,phase-modulated copies of the input optical field

$\begin{matrix}{{{E_{out}(t)} = {{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{E_{in}(t)}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\sum\limits_{n = 1}^{\infty}{r^{n}e^{i\; \beta \; {F_{n}{({\omega_{m}t})}}}{E_{in}\left( {t - {nT}} \right)}}}}}},} & (1)\end{matrix}$

where the parameters k and γ are the coupler power transmission and thepower insertion loss, respectively. The resonator has a FSR of co, atthe input optical frequency ω₀ and roundtrip time T=2π/ω_(r). Thecumulative round-trip field gain is r=√{square root over (α(1−γ)(1−k))},where the light experiences roundtrip power loss (1−α). Ideal, lumpedphase modulation occurs at modulation frequency ω_(m) and modulationindex β. A cascaded modulation function

$\begin{matrix}{{{F_{n}\left( {\omega_{m}t} \right)} = {\sum\limits_{i = 1}^{n}{\sin \; {\omega_{m}\left( {t - {iT}} \right)}}}},} & (2)\end{matrix}$

is defined where the term βF_(n)(ω_(m)t) is the accumulatedtime-dependent phase of the internal field in its nth round trip.Notably, the second term in (1) contains an additional factor of√{square root over (k/(1−k))}.

When the optical input frequency and the modulation frequency areresonant with the FSR (ω₀T and ω_(m)T are integer multiples of 2π,respectively), the output optical field is

$\begin{matrix}{{E_{out}(t)} = {{\sqrt{\left( {1 - \gamma} \right)\; \left( {1 - k} \right)}{\hat{E}}_{i\; n}e^{i\; \omega_{0}t}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}\frac{{re}^{i\; \beta \; \sin \; \omega_{m}t}}{1 - {re}^{i\; \beta \; \sin \; \omega_{m}t}}{\hat{E}}_{i\; n}{\epsilon^{i\; \omega_{0}t}.}}}} & (3)\end{matrix}$

Referring to FIG. 39A, the intra-resonator power spectrum for k=0.03,γ=0, α=0.95, β=π/2, and P_(in)=|E_(in)|²=1 is shown. These values areconsistent with state-of-the-art fabrication technology, and unlessotherwise noted, these are the default parameters used for the rest ofthis disclosure. The intra-resonator optical field is composed ofhundreds of single frequency components spaced at the modulationfrequency, with a large dip at the center frequency. The output comb,however, has a large peak at the input optical frequency due to theunmodulated light that passes through the coupler. The horizontal axisin FIG. 39A is frequency, normalized to the resonator FSR (i.e.,p=ω/ω_(r)). When comb spectra are plotted, the vertical lines thatindicate that the comb is composed of distinct frequency components areomitted, and instead, only the comb envelope is shown.

Referring to FIG. 39B, the intra-resonator temporal power profile insidea RE-EO comb generator is shown, demonstrating pulse formation. Insidethe resonator, phase modulation induces pulse formation when thepreviously mentioned resonant conditions are true. However, because theRE-EO comb generator in FIG. 38 has only a single waveguide, the outputfield contains a superposition of the intra-resonator pulses, as well asthe unmodulated input optical field. While the periodicity of thetemporal power profile is twice the modulation frequency, theperiodicity of the optical field is equal to the modulation frequencydue to the time-dependent phase of the output optical field.

Simplified analytical models have been previously developed for RE-EOcomb generators based on free-space Fabry-Perot resonators but can beadapted to RE-EO comb generators based on ring resonators. For example,the power in the pth comb line of a ring-based RE-EO comb generator isapproximately (β<π, p≠0).

$\begin{matrix}{P_{p} \propto {e^{\frac{{- {p}}{({1 - r^{2}})}}{\beta}}.}} & (4)\end{matrix}$

Increasing the modulation index β and round-trip field gain r results inbroader comb formation.

An analytical solution valid for all cases can be determined when boththe input optical frequency and modulation frequency are resonant. Byapplying a Jacobi-Anger expansion to (1), the output optical field is

$\begin{matrix}{{{E_{out}(t)} = {{\sqrt{\left( {1 - \gamma} \right)\; \left( {1 - k} \right)}{\hat{E}}_{i\; n}\text{?}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\hat{E}}_{i\; n}\text{?}{\sum\limits_{n = 1}^{\infty}r^{n}}}}},{{J_{p}\left( {\beta \; n} \right)}\text{?}},} & (5) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

where J_(p) is the pth order Bessel function of the first kind. Thepower in the pth comb line is then

$\begin{matrix}{\mspace{79mu} {{P_{p} = {\left( {1 - \gamma} \right)\left( {1 - k} \right)\text{?}{{\delta_{p} - {\frac{k}{1 - k}\text{?}r^{n}{J_{p}\left( \beta_{n} \right)}}}}^{2}}},}} & (6) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

where δ_(p) is the Kronecker delta.

Referring to FIG. 40A, the output spectra of a RE-EO comb generator forvarious modulation indices is shown. Comb spectra P_(p) as a function ofcomb line number p for various parameters is shown for differentmodulation indices β. For β>2π, the output comb is highly non-uniformand chaotic. Increasing the modulation index decreases the overall slopeof the comb spectrum. However, for large modulation indices (β>2π), theprofile of the spectrum becomes nonuniform due to the highly oscillatorynature of the Bessel function in (6). Analytical models exist forfree-space RE-EO comb generators driven with large modulation indices,but they will not be discussed further in this disclosure becauseapplications in optical communications require a uniform and flat combspectra.

Referring to FIG. 40B, the output spectra of a RE-EO comb generator isshown. The output comb spectra as a function of comb line number p isshown for different coupling power transmissions k for the sameparameters as FIG. 40A. Since the intra-resonator power loss (1−α) andthe coupler insertion loss γ are often fixed, the coupler powertransmission k can be tuned to change the output spectrum. As k isdecreased, the comb slope decreases and the comb flatness increasesbecause the intra-resonator field is modulated during more round tripsbefore being output-coupled. However, decreasing k also decreases theefficiency η=Σ_(p≠0) P_(p)/P_(in) of the comb generation process becausemore of the input optical field is passed directly through the couplerand into the output waveguide.

Output Noise

1) Optical Input Phase Noise:

Previously, it was assumed that the input optical field contained asingle frequency. This assumption is now relaxed, and it is assumed thatthe input field is E_(in)(t)=Ê_(in)e^(iω) ⁰ ^(t+iθ) ⁰ ^((t)) where θ₀(t)is the phase noise of the input optical field. The power spectraldensity (PSD) of E_(in)(t) is

$\begin{matrix}\begin{matrix}{{S_{i\; n}(\omega)} = {\int_{- \infty}^{\infty}{{\langle{{E_{i\; n}(t)}{E_{i\; n}^{*}\left( {t + \tau} \right)}}\rangle}e^{{- i}\; \omega \; \tau}d\; r}}} \\{{= {P_{i\; n}{\int_{- \infty}^{\infty}{{\langle e^{i\; \Delta_{{\gamma\theta}_{0}}}\rangle}e^{- {i{({\omega + \omega_{\theta}})}}^{\tau}}d\; r}}}},}\end{matrix} & (7)\end{matrix}$

where <.> denotes averaging over time t, and Δ_(τ)θ₀=θ₀(t+τ)−θ₀(t).

When both the input optical frequency and the modulation frequency areresonant, the output PSD of the pth comb line in the presence of inputoptical phase noise is

$\begin{matrix}{{{S_{{out},p}(\omega)} = {k^{2}\frac{1 - \gamma}{1 - k}{{\chi_{p}(\omega)}}^{2}{S_{i\; n}\left( {\omega - {p\; \omega_{m}}} \right)}}},} & (8)\end{matrix}$

where the frequency-dependent linewidth correction term for the pth combline is

$\begin{matrix}{{\chi_{p}(\omega)} = {\sum\limits_{n = 1}^{\infty}{r^{n}{J_{p}\left( \beta_{n} \right)}{e^{{- i}\; \omega \; {nT}}.}}}} & (9)\end{matrix}$

From (8), it is evident that multiplication by |χ_(p)(ω)|² changes theshape of the PSD of the pth comb line phase noise. The calculations toderive this result are detailed below.

Referring to FIG. 41, plots of the normalized linewidth correctionfactor |χ_(p)(ω)|² for various comb line numbers p are shown. Because|χ_(p)(ω)|² is locally flat for co inside the input optical linewidth,the input optical phase noise is approximately copied to each of thecomb lines. The horizontal axis is frequency from the center of the pthcomb line, normalized to the resonator FSR. For many practical ringresonators, the FSR can be over four orders of magnitude larger than thelinewidth of the input optical carrier. For this reason, only the shapeof |χ_(p)(ω)|² near ω=ω₀A+pω_(m) contributes to a change in linewidth.From the vertical scale of FIG. 41 it is clear that the linewidthcorrection term is locally flat for frequencies within the linewidth ofthe input optical field. For this reason, the phase noise of the pthcomb line is nearly identical to the phase noise of the input opticalfield.

2) Modulation Phase Noise:

Similar to the above, the impact of modulator phase noise can beanalyzed by introducing a time-dependent phase θ_(e)(t) into the phasemodulation. The cascaded modulation function, including modulation phasenoise is

$\begin{matrix}\begin{matrix}{{F_{n}\left( {\omega_{m}t} \right)} = {\sum\limits_{i = 1}^{n}{\sin \left\lbrack {{\omega_{m}\left( {t - {iT}} \right)} + {\theta_{e}\left( {t - {iT}} \right)}} \right\rbrack}}} \\{{\approx {n\; {\sin \left\lbrack {{\omega_{m}t} + {\theta_{e}(t)}} \right\rbrack}}},}\end{matrix} & (10)\end{matrix}$

where it is assumed that θ_(e)(t) is slowly varying over relevantresonator time scales such as the resonator decay lifetime. This is asafe assumption because crystal-controlled microwave oscillators usedfor modulation have coherence times much longer than those of opticalresonators. By inserting (10) into (1) and assuming resonance of theinput optical frequency and modulation frequency, the output field ofthe pth comb line (p≠0) is

$\begin{matrix}{\mspace{79mu} {{E_{p}(t)} = {{- k}\sqrt{\frac{1 - \gamma}{1 - k}}{{\hat{E}}_{i\; n}\left\lbrack {\sum\limits_{n = 1}^{\infty}{r^{n}{J_{p}\left( {\beta \; n} \right)}}} \right\rbrack} \times {\text{?}.}}}} & (11) \\{\text{?}\text{indicates text missing or illegible when filed}} & \;\end{matrix}$

The PSD of the pth comb line in the presence of modulation phase noiseis then

S _(out,p)(ω)=P _(p)∫_(−∞) ^(∞)

e ^(ipΔ) ^(τ) ^(θ) ^(e)

e ^(−iωτ) dτ  (12)

where Pp is given by (6) and Δ_(τ)θ_(e)=θ_(e)(t+τ)−θ_(e)(t). If it isassumed that Δ_(τ)θ_(e) is a Gaussian random process, then the linewidthof the pth optical field, Δω_(p), is related to the

phase noise by Δω_(p)|τ|=

(pΔ _(τ)θ_(e))²

.  (13)

From this relation, it is clear that the linewidth of the pth comb lineincreases quadratically with p. The quadratic dependence of thelinewidth on comb line number can introduce significant noise forapplications that require thousands of comb lines, such as precisiontiming. However, for applications that require hundreds of comb lines orless, the output phase noise is still dominated by input laser phasenoise. High-frequency comb line phase noise can be filtered by inputtingan optical frequency slightly detuned away from a harmonic of the FSR.This effect results from a frequency dependent filtering term in (11),though a detailed analysis is not presented here.

It was assumed above that both the input optical frequency and themodulation frequency were harmonics of the resonator FSR. In practicalsystems, this assumption is not always satisfied. In order to maintainthis resonance condition, various locking methods may be used to ensurethat the desired comb properties are preserved. Here, since the mostimportant comb property for optical communications is comb power,impairments to the output spectrum in the presence of optical frequencyoffsets and modulation frequency offsets are analyzed.

Non-Resonant Optical Input

It is first assumed that the input field has an optical frequency offsetΔω₀ such that the input optical field is E_(in)(t)=Ê_(in)e^(i(ω) ⁰^(+Δω) ⁰ ^()t). The normalized optical frequency offset ϕ₀=Δω_(o)T isdefined. From (1), the output power in the pth comb line in the presenceof an input optical frequency offset is

$\begin{matrix}{P_{p,o} = {k^{2}\frac{1 - \gamma}{1 - k}P_{i\; n}{{{\sum\limits_{n = 1}^{\infty}{\left( {re}^{{- i}\; \varphi_{o}} \right)^{n}{J_{p}\left( {\beta \; n} \right)}}}}^{2}.}}} & (14)\end{matrix}$

FIG. 42 illustrates output spectra from a non-resonant RE-EO combgenerator. FIG. 42A is a plot of power spectra P_(p,o) for differentoptical frequency offsets. For ϕ_(o)>β, the comb width is reducedsubstantially. FIG. 42B is a plot of power spectra P_(p,m) for differentmodulation frequency offsets, demonstrating reduction of comb width forlarge ϕ_(m).

FIG. 42A shows the output comb spectrum calculated from (14) for variousvalues of ϕ₀. For small values of ϕ₀, the shape of the comb remainsunchanged, while for values of ϕ₀ that approach and surpass themodulation index β, the comb drastically decreases in size. The absolutephase of each of the comb lines also changes, along with the opticalpower, but will not be discussed further here. For unmodulatedresonators, inputting an optical frequency inside of the resonatorlinewidth Δω_(r)=(1−r²)ω_(r)/(2π) is critical to increasing the built-uppower in the resonator. Thus, one might expect that inputting an opticalfrequency outside of Δω_(r) into a modulated resonator may reduce thecomb generation efficiency. Indeed for some values of ϕ₀, such asϕ₀>0.57π, the output comb is much less flat. However, flat combs can begenerated even in the presence of normalized optical frequency offsetsgreater than the normalized resonator linewidth, ϕ_(r)=Δω_(r)T=(1−r²).For the default parameters assumed in this paper, ϕ_(r)=0.0137π, whilebroad comb formation is evident for ϕ₀=0.257π in FIG. 42A. These resultsagree with previous experimental results from free-space REEO combgenerators.

Non-Resonant Modulation

It is now assumed that the modulator is driven with modulation frequencyoffset Δω_(m) and define the normalized modulation frequency offsetϕ_(m)=Δω_(m)T. From (1), the power in the pth comb line in the presenceof a modulation frequency offset is

$\begin{matrix}{{P_{p,m} = {\frac{\left( {1 - \gamma} \right)}{\left( {1 - k} \right)}k^{2}P_{i\; n} \times {{\sum\limits_{n = 1}^{\infty}{\sum\limits_{q = {- \infty}}^{\infty}{r^{n}i^{q}{J_{p - q}\left( {\beta_{o}\left( {\varphi_{m},n} \right)} \right)}{J_{q}\left( {\beta_{e}\left( {\varphi_{m},n} \right)} \right)}}}}}^{2}}},} & (15)\end{matrix}$

where the modified odd and even modulation indices β₀(ϕ_(m), n) andβ_(e)(ϕ_(m), n) are defined as

$\begin{matrix}{{\beta_{o}\left( {\varphi_{m},n} \right)} = {\beta\left( {{\frac{1}{2}{\cot \left( {\varphi_{m}/2} \right)}} - \frac{\cos \left( {\left( {n + \frac{1}{2}} \right)\varphi_{m}} \right)}{2\; {\sin \left( {\varphi_{m}/2} \right)}}} \right)}} & (16) \\{{\beta_{e}\left( {\varphi_{m},n} \right)} = {{\beta\left( {{- \frac{1}{2}} + \frac{\sin \left( {\left( {n + \frac{1}{2}} \right)\varphi_{m}} \right)}{2\; {\sin \left( {\varphi_{m}/2} \right)}}} \right)}.}} & (17)\end{matrix}$

The calculations to derive (15) from (1) are included in Appendix B.Since (15) introduces an additional infinite summation, the complexityof the calculation increases significantly, especially in cases wherethe resonator loss is small or modulation index is large. An efficientnumerical model is provided that approximates this analytical model.FIG. 42B shows comb spectra in the presence of various modulationfrequency offsets. For small modulation frequency offsets, the combremains flat. An increase in φ_(m), however, leads to an inverselyproportional decrease of the total width from approximately 120 comblines to 40 comb lines due to an increase of the modulation frequencyoffset from ϕ_(m)=0.017π to ϕ_(m)=0.037π. As above, the comb spectra inFIG. 42 exhibit behavior that deviates from the behavior of anunmodulated resonator. Comb lines with frequencies that lie far outsideof the resonator linewidth still build up in the resonator. For example,for ϕ_(m)=0.037π, all of the approximately 40 generated comb frequencieslie outside of the normalized resonator linewidth, ϕ_(r)=0.0137π. Largering resonators, corresponding to large T, are much more sensitive tooptical frequency offsets and modulation frequency offsets because thenormalized frequency offsets are linearly dependent on the resonatorlength. For this reason, it is ideal for the modulation frequency toequal the fundamental FSR rather than a harmonic of the FSR in order toincrease the tolerable frequency offsets.

While the analytical models above exactly predict the output combspectra of a RE-EO comb generator in resonant and non-resonantoperation, these models cannot include arbitrary frequency-dependenteffects such as dispersion. Two methods of numerically approximating theoutput spectrum of a dispersive RE-EO comb generator are provided.

A. Round-Trip Phase Model

An intuitive understanding of the resonance conditions of an RE-EO combgenerator can be approached first from the resonance conditions of anunmodulated resonator, which can be fully explained through theinterference of internal and external fields. For example, a typicalresonance condition for an input optical field with frequency ω_(p)coupled to a resonator with normalized linewidth ϕ_(r), as definedabove, is

|θ_(p,tot)|<ϕ_(r)/2,  (18)

where θ_(p,tot)=ω_(p)T mod 2π is the total round-trip accumulated phaseoffset of the optical field. Frequencies that do not satisfy thiscondition do not experience constructive interference inside theresonator. However, the intra-resonator phase modulation introduces atime-dependent variation in the resonance condition that results inconstructive interference at one or more locations inside the resonator,depending on whether the phase modulation is equal to, or a subharmonicof, the FSR. As a result of this spatially varying constructiveinterference, intra-resonator pulses are formed. The new condition forconstructive interference in the resonator is |θ_(p,tot)+β sinω_(m)t|<ϕ_(r)/2. Since this condition may be satisfied for any time tthe resonance condition becomes

−β<θ_(p,tot)<β,  (19)

where the finite resonator linewidth is omitted because it is often muchsmaller than the modulation index. This resonance condition explains thecomb formation effects, where comb lines were generated even though theywere outside of the resonator linewidth. To validate this model,consider a RE-EO comb generator that is modulated exactly at theresonator FSR (ϕ_(m)=0) but has some known optical frequency offset ϕ₀.In the absence of dispersion, the round-trip accumulated phase of thepth comb line is ϕ_(p,tot)=ϕ₀. For unmodulated resonators, constructiveinterference inside the resonator can easily be verified by changing theoptical frequency offset and measuring a dip in the transmissionspectrum. Analogous to (3), the timedependent output field in thepresence of optical frequency offset is

$\begin{matrix}{{E_{out}(t)} = {{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{\hat{E}}_{i\; n}e^{i\; \omega_{0}t}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}\frac{\left( {re}^{{- i}\; \varphi_{o}} \right)e^{i\; \beta \; \sin \; \omega_{m}t}}{1 - {\left( {re}^{{- i}\; \varphi_{o}} \right)e^{i\; \beta \; \sin \; \omega_{m}t}}}{\hat{E}}_{i\; n}{e^{i\; \omega_{0}t}.}}}} & (20)\end{matrix}$

FIG. 43 illustrates time-averaged power transmission

|E_(out)(t)|²

for various optical frequency offsets ϕ₀ and modulation indices β.Dashed lines correspond to the edge of the round-trip phase modelresonance condition as described herein.

FIG. 43 shows the time-averaged output power

|E_(out)(t)|²

as a function of the normalized optical frequency offset for variousmodulation indices. A narrow dip is observed in the power transmissionfor β=0, corresponding to the expected output from a unmodulatedresonator. For β≠0, however, constructive interference occurs at manyvalues of the optical frequency offset. The dashed lines in FIG. 43correspond to the limits of the round-trip phase model (−β<ϕ₀<β), whichaccurately predict the maximum optical frequency offset that results inintra-resonator power build-up. The round-trip phase model can beextended to include effects such as modulation frequency offsets as wellas dispersion. In a single round trip, the accumulated phase of the pthcomb line due to a modulation frequency offset, θ_(p,m), is linear incomb line number, i.e.,

θ_(p,m)=[ω₀ T+p(ω_(m)+Δω_(m))T]mod 2π=pϕ _(m),  (21)

The effects of dispersion can also be included by integrating themeasured or simulated group velocity dispersion to determine theround-trip accumulated phase offset of the pth comb line due todispersion, θ_(p,d). However, if it is assumed that a linear dispersionprofile, θ_(p,d), is

θ_(p,m)=[ω₀ T+p(ω_(m)+Δω_(m))T]mod 2π=pϕ _(m),  (21)

where β₂L is the round-trip group velocity dispersion and the normalizedphase offset due to dispersion is then ϕ_(d)=ω_(m) ²β₂L. Finally, theresonance condition for an RE-EO comb generator including opticalfrequency offsets, modulation frequency offsets, and linear dispersionis

−β<ϕ_(o) +pϕ _(m) +p ²ϕ_(d)<β.  (23)

Similar expressions can be extracted and the dispersion-limited combwidth of free-space REEO comb generators can be analyzed usingFabry-Pérot resonators. For a linear dispersion profile, the maximumcomb width occurs when ϕ₀=−β and is given by

${\Delta \; \omega_{comb}} = {\sqrt{\frac{2\; \beta}{\beta_{2}L}}.}$

This value agrees with previous comb widths up to a factor of √{squareroot over (2)} due to the difference in FSR of a Fabry-Pérot resonatorand ring resonator of identical length. To fully characterize the outputpower spectrum, the following assumptions are made: (a) the light in thecenter frequency is dominated by the input field that passes through thecoupler, i.e., P₀=(1−γ)(1−k)P_(in), (b) the slope of the comb spectrumis given by (4), and (c) the power in the first sideband is given by

$\begin{matrix}{{P_{\pm 1} = {\frac{k^{2}\left( {1 - \gamma} \right)}{1 - k}{J_{1}(\beta)}^{2}}},} & \;\end{matrix}$

simplified from (6). These assumptions, along with (23) form theround-trip phase model, which can efficiently predict the approximateshape of the output comb spectrum. The round-trip phase model as well asthe modeling above have been successfully used to predict the outputcomb spectrum of actual integrated ring structures.

B. Steady-State Matrix Method

One drawback of the round-trip phase model is that fine features of theoutput comb spectrum cannot be determined, as shown by variations incomb line power of up to 10 dB in FIG. 42B. A numerical method isprovided that is capable of resolving these fine features by determininga relation between the comb lines inside the resonator. Similarcalculations have been performed before in the context of freespaceFabry-Pérot resonators, but not ring-based cavities. First, theintra-resonator field E_(c)(t), shown in FIG. 38, is propogated by oneround-trip time T resulting in the following relation:

E _(c)(t)=√{square root over (α)}e ^(iβ sin ω) ^(m) ^(t)[√{square rootover ((1−γ)(1−k))}E _(c)(t+T)+i√{square root over ((1−γ)k)}E_(in)(t+T)].  (24)

It is assumed that that E_(c)(t) can be expressed as a superposition ofoptical fields with frequencies spaced at the modulation frequency,i.e.,

$\begin{matrix}{{{E_{c}(t)} = {\sum\limits_{p = {- \infty}}^{\infty}{E_{p}e^{{i{({\omega_{0} + {p\; \omega_{m}}})}}^{t}}}}},} & (25)\end{matrix}$

where E_(p) is the complex optical field of the pth comb line inside theresonator. If the optical field has reached steady state, correspondingto many round trips after the light is first input-coupled into theresonator, the relation between all E_(p) is

$\begin{matrix}{{E_{p} = {{r{\underset{q = {- \infty}}{\sum\limits^{\infty}}{{J_{q}(\beta)}E_{p - q}e^{i\; \theta_{{p - q},{tot}}}}}} + {i\sqrt{\frac{k}{1 - k}}r{\hat{E}}_{i\; n}{J_{p}(\beta)}e^{i\; \theta_{0,{tot}}}}}},} & (26)\end{matrix}$

where θ_(p,tot)=ϕ₀+θ_(p,m)+θ_(M) is the round-trip normalized frequencyoffset of the pth comb line. This system of linear equations can besolved with simple matrix methods. The output field in the waveguide is

E _(out)(t)=√{square root over ((1−γ)(1−k))}E _(in)(t)+i√{square rootover ((1−γ)k)}E _(c)(t),  (27)

where the values of the complex optical field E_(p) are solved above.

In practice, when using a matrix solver to compute E_(p), it isnecessary to increase the number of simulated comb lines because themodel may become inaccurate at the edges of the spectrum. This effectoccurs because frequency conversion from carriers outside of the widthof the simulation are not included. Since this method is quiteefficient, increasing the number of simulated comb lines by even afactor of two is often tolerable. Although not discussed further in thispaper, these equations reveal individual phase information of the comblines, which have analytical solutions, and may be useful forapplications where relative phase information is desired.

C. Comparison of Methods

Here, the three models of computing the output spectrum arevalidated—the analytical model, the round-trip phase model, and thesteady-state matrix method—by comparing the predicted output spectra inthe presence of modulation frequency offsets. Since optical frequencyoffsets solely change the slope of the comb, this comparison is omitted.

FIG. 44 provides a comparison of methods used to predict the output combspectrum. FIG. 44A is a plot of output comb spectra P_(p) calculatedwith the analytical model (lines) and the round-trip phase model (filledcircles) for various modulation frequency offsets. FIG. 44A is a plot ofoutput comb spectra P_(p) calculated with the analytical model (lines)and the steady-state matrix method (filled circles) for variousmodulation frequency offsets.

FIG. 44A shows the spectra calculated from the analytical model (lines)and the round-trip phase model (filled circles) for various modulationfrequency offsets.

The round-trip phase model accurately predicts the comb width and shape,but fails to predict the fine features of the comb spectrum, asexpected. FIG. 44B compares the spectra computed with the analyticalmodel (lines) and the steady-state matrix method (filled circles). Here,the steady-state matrix method is able to calculate the fine features ofthe comb spectrum with high accuracy.

The round-trip phase model and the steady-state matrix method can alsobe used to predict the effects of dispersion. In the following, a lineardispersion profile is assumed (i.e., θ_(p,d)=p²ϕ_(d)) withϕ_(d)=2π×10⁻⁴. This value of ϕ_(d) is considerably larger than anypractical values in order to emphasize the effects of dispersion. Forexample, for a lithium niobate resonator with 10 GHz FSR, this value ofϕ_(d) would correspond to a group velocity dispersion of ˜1.2×10⁴fs2/mm, over two orders of magnitude larger than that of currentwaveguide technology.

FIG. 45 shows output comb spectra P_(p) from the round-trip phase model(dotted line) and the steady-state matrix method (solid line). In thepresence of modulation frequency offsets and dispersion, complex andasymmetric comb spectra can be observed.

FIG. 45 compares output comb spectra for various modulation frequencyoffsets for a linear dispersion profile. As in FIG. 44, the round-tripphase model is able to predict the comb shape, but is not able todetermine fine comb features.

Conversely, the steady-state matrix method is able to resolve finefeatures in the comb spectra.

When both modulation frequency offsets and dispersion are included, thecomb spectra becomes asymmetric about the center frequency. This effectresults from the resonance condition −β<pϕ_(m)+p²ϕ_(d)<β, where theresonance condition for positive and negative p is different. For higherfrequency comb lines (p>0), both the modulation frequency offset and thedispersion phase offset have the same sign while for lower-frequencycomb lines (p<0), they have opposite signs. Unlike many other combgenerators, such as those based on χ⁽³⁾ nonlinear effects, the RE-EOcomb generator does not require extensive dispersion engineering toproduce viable frequency combs because it does not require phasematching over long periods of time.

As mentioned above, resonator-based comb generators often have lowefficiency due to low coupling between the input waveguide andresonator. A frequency-dependent coupler with high transmission at theinput frequency, but low transmission at all other frequencies can solvethis problem because the input light can efficiently coupled into theresonator where the newly generated frequencies may then resonate formany round trips. While complicated frequency dependent couplers basedon photonic crystals or distributed Bragg reflectors can be fabricatedto approach the desired frequency response, these methods introduceadditional fabrication requirements and excess insertion loss. Theimpact on the output spectrum of an additional ring coupler used toefficiently couple the input field to the comb-generating resonator isanalyzed.

FIG. 46 is a diagram of a dual-ring EO comb generator. Power at theinput optical frequency builds up in the small ring and is then coupledto a larger resonator, where the comb is generated. A third coupler isused to output-couple the desired comb.

FIG. 46 illustrates a dual-ring EO comb generator according toembodiments of the present disclosure. An input optical field E_(in)(t)is first resonantly coupled into a small ring. Inside that ring, powerbuilds up at the input optical frequency, before it is coupled into thelarger, comb-generating resonator. This resonator includes a coupler tooutput the desired frequency comb. The coupler power transmissioncoefficients are k1, k2, and k3 and the coupler insertion losses are γ₁,γ₂, γ₃, as shown in FIG. 46. The power losses of the small ring,(1−{tilde over (α)}), and of the combgenerating resonator, (1−α), arerelated by {tilde over (α)}=α{circumflex over ( )}({tilde over(T)}/T),where 1/{tilde over (T)} and 1/T are the FSRs of the small ringand combgenerating resonator, respectively. The new output fieldE_(out)(t) can be calculated as a function of the input field E_(in)(t)with the steady-state matrix method. However, the small ring couplerintroduces frequency dependent phase shifts, loss, and powertransmission. To determine a self-consistent relation between the combline optical fields E_(p), analogous to (26) that includes theseeffects, the same techniques as above can be applied.

First, the field in the small resonator, {tilde over (E)}_(c)(t), andthe field in the larger resonator, E_(c)(t), can be related by thefollowing equations:

$\begin{matrix}{{{\overset{\sim}{E}}_{c}(t)} = {{{\overset{\sim}{r}}^{\prime}{{\overset{\sim}{E}}_{c}\left( {t + \overset{\sim}{T}} \right)}} + {i\sqrt{\frac{k_{2}}{1 - k_{2}}}\left( \frac{\overset{\sim}{\alpha}}{\alpha} \right)^{\frac{1}{4}}r^{\prime}{E_{c}\left( {t + {\overset{\sim}{T}/2} + {T/2}} \right)}} + {i\sqrt{\frac{k_{1}}{1 - k_{1}}}{\overset{\sim}{r}}^{\prime}{E_{i\; n}\left( {t + \overset{\sim}{T}} \right)}}}} & (28) \\{{{E_{c}(t)} = {{r^{\prime}e^{i\; \beta \; \sin \; \omega_{m}t}{E_{c}\left( {t + T} \right)}} + {i\sqrt{\frac{k_{2}}{1 - k_{2}}}\left( \frac{\alpha}{\overset{\sim}{\alpha}} \right)^{\frac{1}{4}}{\overset{\sim}{r}}^{\prime}e^{i\; \beta \; \sin \; \omega_{m}t}{{\overset{\sim}{E}}_{c}\left( {t + {\overset{\sim}{T}/2} + {T/2}} \right)}} - {\left( {\alpha \; \overset{\sim}{\alpha}} \right)^{\frac{1}{4}}\sqrt{\left( {1 - \gamma_{1}} \right){k_{1}\left( {1 - \gamma_{2}} \right)}k_{2}}e^{i\; \beta \; \sin \; \omega_{m}t} \times {E_{i\; n}\left( {t + {\overset{\sim}{T}/2} + {T/2}} \right)}}}},} & (29)\end{matrix}$

where {tilde over (r)}′=√{square root over ({tilde over(α)}(1−γ₁)(1−k₁)(1−γ₂)(1−k₂))} is the round-trip gain coefficient of thesmall ring and r′=√{square root over (α(1−γ₂)(1−k₂)(1−γ₃)(1−k₃))} is theround-trip gain coefficient of the comb-generating resonator. If it isassumed that both E_(c)(t) and {tilde over (E)}_(c)(t) aresuperpositions of fields spaced at the modulation frequency, analogousto (25), the complex field of the pth comb line, E_(p), is related tothe other comb fields and input field via the following expression:

$\begin{matrix}{{E_{p} = {{\sum\limits_{p = {- \infty}}^{\infty}{r^{\prime}{J_{q}(\beta)}{e^{i\; \omega_{p - q}T}\left( \frac{1 - {{\overset{\sim}{r}}^{\prime}{e^{i\; \omega_{p - q}\overset{\sim}{T}}/\left( {1 - k_{2}} \right)}}}{1 - {{\overset{\sim}{r}}^{\prime}e^{i\; \omega_{p - q}\overset{\sim}{T}}}} \right)}E_{p - q}}} - {\left( {\alpha \; \overset{\sim}{\alpha}} \right)^{1/4}\sqrt{\left( {1 - \gamma_{1}} \right){k_{1}\left( {1 - \gamma_{2}} \right)}k_{2}} \times {J_{p}(\beta)}{e^{i\; {\omega_{0}{({{T/2} + {\overset{\sim}{T}/2}})}}}\left( \frac{1}{1 - {{\overset{\sim}{r}}^{\prime}e^{i\; \omega_{0}\overset{\sim}{T}}}} \right)}}}},} & (30)\end{matrix}$

where ω_(p)T=θ_(p,tot) is the accumulated round-trip phase of the pthcomb line in the comb-generating resonator and ω_(p){tilde over(T)}=θ_(p,tot)({tilde over (T)}/T)+pω_(m){tilde over (T)} is theround-trip accumulated phase of the pth comb line in the small ring. Theoutput optical field E_(out)(t) and the reflected field E_(r)(t) are

E _(out)(t)=i√{square root over ((1−γ₃)k ₃)}E _(c)(t)  (31),

and

E _(r)(t)=i√{square root over ((1−γ₁)k ₁)}{tilde over (E)}_(c)(t)+√{square root over ((1−γ₁)(1−k ₁))}E _(in)(t).  (32)

With these expressions, a simple matrix solver may be used to firstcalculate E_(c)(t) and then find the output field E_(out)(t).

It is assumed that α=0.95, β=/2, and P_(in)=|Ê_(in)|²=1. Additionally,to provide a fair comparison to the single resonator comb generator, itis assumed that γ₁=γ₂=γ₃=0 and k_(i)=k₂=k₃=0.03. Finally, {tilde over(T)}=T/50 is chosen in order to prioritize the 100 comb lines closest tothe center optical frequency.

FIG. 47 is a plot of the output spectra P_(p) of a single-ring combgenerator and a dual-ring comb generator. The reflected field from thedual-ring EO comb generator is also shown, demonstrating constructivebuild-up at the FSR of the coupling ring.

FIG. 47 compares the output comb spectrum of a single ring RE-EO combgenerator to the output and reflected comb spectra of a dual-ring RE-EOcomb generator. As expected, the small ring acts as afrequency-dependent coupler, with high transmission at multiples of thesmall ring FSR, but low transmission at other frequencies. This effectis demonstrated by the reflected power spectrum in FIG. 47, where poweris concentrated in frequencies at a multiple of the small ring FSR. Theoutput spectrum from the dual-ring RE-EO comb generator is significantlyhigher for comb lines that are within the FSR of the small ring. Theconversion efficiency of input power to the 100 nearest comb linesexcluding the center line (−50<p<50, p≠0 for the single-ring combgenerator is 1.3%, while the conversion efficiency of the dual-ring combgenerator is 32.1%. The average increase in output power of these comblines is 13.9 dB. However a sharp dip in comb line power occurs atmultiples of the small ring FSR because light at these frequencies iscoupled efficiently back into the smaller ring.

One possible limitation to the dual-ring comb generator design is thematerial damage threshold of the small ring. For the parametersdiscussed above, the time-averaged power in the small ring is 27 timesthe input optical power. However, even if the input optical power isunrealistically high, such as 1 W, the intra-resonator power is a factorof two below the damage threshold of many state-of-the-art integratedresonators.

In some cases, fabricating a ring with a FSR that is 50 times higherthan that of the desired comb line spacing may prove challenging due tofabrication or material constraints. In these cases, it is stillpossible to generate high-power combs by increasing the size of the ringcoupler, but tuning its length so that its FSR is not a harmonic of theFSR of the comb generating resonator.

FIG. 48 is a plot of output spectra P_(p) for different values of theratio of FSRs, T/{tilde over (T)}, of the small ring and comb-generatingresonator. Slightly tuning the FSR of the coupling ring away from aharmonic of the FSR of the comb-generating resonator can increase thecomb width and power.

FIG. 48 shows the output spectrum of a dual-ring EO comb generator forvarious values of the ratio of FSRs, T/{tilde over (T)}, of the smallring and comb-generating resonator. When compared to a small ringcoupler (T/{tilde over (T)}=50), a ring coupler of increased size withFSR at a harmonic of the combgenerating resonator (T/{tilde over(T)}=10) results in a narrower comb due to interference inside the ringcoupler. However, if the FSR of the small ring is adjusted away from aharmonic of the comb-generating resonator, the total FSR of the systemincreases, resulting in broad, high power comb generation without thedifficulty of fabricating ultra-small rings. This process cannot proceedindefinitely due to the finite linewidth of the small ring resonator.Once the finesse of the ring approaches the ratio of the FSRs, at leastone comb line inside the comb-generating resonator will lie inside thelinewidth of the small ring. In this case, that comb line will becoupled back into the small ring and the FSR of the entire system islimited by the small ring FSR.

As mentioned above, frequency combs can be used in WDM coherent opticalcommunications systems for both the transmitted optical carrier andreceiver local oscillator. One problem with single-ring RE-EO combgenerators is the low output power in each of the comb lines, whichlimits the OSNR of the transmitted optical carriers. The OSNR of WDMoptical links utilizing RE-EO comb generators is analyzed.

FIG. 49 is a schematic view of a WDM point-to-point inter-data centerlink. The output field from the RE-EO comb generator is flattened,amplified, and de-multiplexed (Demux). Each of the comb lines ismodulated (Mod), multiplexed (Mux), and amplified before being input toa link of single-mode fiber (SMF). At the receiving end, the signal isamplified and sent to a coherent receiver (Rx). The OSNR is measured atthe receiver input.

FIG. 49 shows an example WDM link that utilizes a RE-EO comb generatorthat seeds each of the modulated comb frequencies. An input laser iscoupled into a RE-EO comb generator where multiple comb lines aregenerated. The output comb is flattened, amplified with a boosteramplifier, and sent to a (de-)multiplexing stage, where the comb linesare separated, modulated individually, and re-combined. After thisstage, the comb lines are amplified, transmitted through a length ofsingle-mode fiber (SMF), and amplified before being sent to a coherentreceiver.

Table 1 lists the parameters for this calculation. Notably, thedifference in booster amplifier power between systems that utilize asingle-ring and dual-ring RE-EO comb generator is 10 dB. Thisperformance improvement is smaller than the average comb line powerimprovement of 14 dB because nonuniformities in the dual-ring RE-EO combgenerator output spectrum result in a lower minimum optical power thanthe single-ring RE-EO comb generator. This effect is evident in FIG. 47.

TABLE 1 TABLE I WDM LINK PARAMETERS Input laser power 20 dBm Insertionloss from output-coupling and flattening  5 dB Booster amplifier noisefigure  5 dB Booster amplifier gain* 30 dB Insertion loss from(de-)multiplexing and modulation 20 dB Link amplifier noise figure  5 dBLink amplifier gain 20 dB Insertion loss from SMF 20 dB Local oscillatorpower 15 dBm *20 dB for dual-ring RE-EO comb generator

For the values listed above, the receiver-side OSNR for a single-ringRE-EO comb generator is 21 dB, while the receiver-side OSNR for thedual-ring design is 28 dB. For a typical 28 Gbaud dual-polarization linkbased on 16-array quadrature amplitude modulation, the requiredreceiver-side OSNR is ˜22 dB [44]. For a WDM link that employs 100 comblines, as shown in FIG. 47, the total bit-rate per fiber for thissystem, including overhead is 20 Tb/s. While single ring RE-EO combgenerators can support lower modulation formats, and thus lowerbit-rates per fiber, dual-ring comb generators provide an increase inOSNR budget of over 7 dB.

Analytical and numerical methods of predicting the output comb spectrumin the presence of a variety of impairments including optical frequencyoffsets, modulation frequency offsets, and dispersion are provided.These models are validated against each other and demonstrate thatnumerical modeling can efficiently approximate the comb spectrum withoutsacrificing accuracy. However, RE-EO comb generators based on a singleresonator often cannot generate enough comb power to be useful forapplications such as optical communications. Thus a fabricable RE-EOcomb generator design is provided that utilizes a ring coupler toenhance the efficiency of the comb generation process. For this newdesign, the conversion efficiency is 30% higher than designs based on asingle resonator, which enable its use in high-capacity coherent opticalcommunications systems.

Output Phase Noise

This appendix calculates the relation between the phase noise of the pthcomb line and the phase noise of the input optical field for resonantoperation, as discussed above. It is assumed that the input opticalfield is E_(in)(t)=Ê_(in)e^(iω) ⁰ ^(t+iω) ⁰ ^((t)), with PSD, S_(in)(ω),given by (7). The PSD of the output field, E_(out)(t) is

$\begin{matrix}\begin{matrix}{{S_{out}(\omega)} = {\int_{- \infty}^{\infty}{{\langle{{E_{out}(t)}{E_{out}^{*}\left( {t + \tau} \right)}}\rangle}e^{{- i}\; \omega \; \tau}d\; \tau}}} \\{= {\left( {1 - \gamma} \right)\left( {1 - k} \right){S_{i\; n}\left( {\omega - {\left( {1 - \gamma} \right){{kS}_{o,1}(\omega)}} -} \right.}}} \\{{{{\left( {1 - \gamma} \right){{kS}_{o,2}(\omega)}} + {\left( {1 - \gamma} \right)\frac{k^{2}}{1 - k}{S_{o,3}(\omega)}}},}}\end{matrix} & (33)\end{matrix}$

where S_(o,1)(ω), S_(o,2) (ω), and S_(o,3) (ω), result from theautocorrelation of E_(out)(t), given by (5), and are

$\begin{matrix}{{S_{o,1}(\omega)} = {\int_{- \infty}^{\infty}{{\langle{{E_{i\; n}(t)}{\sum\limits_{n = 1}^{\infty}{r^{n}e^{{- i}\; \beta \; n\; \sin \; {\omega_{m}{({t + \tau})}}} \times {E_{i\; n}^{*}\left( {t + \tau - {nT}} \right)}}}}\rangle}e^{{- i}\; \omega \; \tau}d\; \tau}}} & (34) \\{\mspace{79mu} {{S_{o,2}(\omega)} = {\int_{- \infty}^{\infty}{{\langle{{E_{i\; n}^{*}\left( {t + \tau} \right)}{\sum\limits_{n = 1}^{\infty}{r^{n}e^{i\; \beta \; n\; \sin \; \omega_{m}t} \times {E_{i\; n}\left( {t - {nT}} \right)}}}}\rangle}e^{{- i}\; \omega \; \tau}d\; \tau}}}} & (35) \\{{S_{o,3}(\omega)} = {\int_{- \infty}^{\infty}{{\langle{\sum\limits_{n,{m = 1}}^{\infty}{r^{n + m}e^{i\; \beta \; n\; \sin \; \omega_{m}t}e^{{{- i}\; \beta \; m\; \sin \; {\omega_{m}{({t + \tau})}}}\;} \times {E_{i\; n}\left( {t - {nT}} \right)}{E_{i\; n}^{*}\left( {t + \tau - {mT}} \right)}}}\rangle}e^{{- i}\; \omega \; \tau}d\; {\tau.}}}} & (36)\end{matrix}$

Focusing first on S_(o,1)(ω), and S_(o,2) (ω), the optical phase noiseis uncorrelated to the phase modulation and thus the time-averaginginside the integrals can be separated into two terms, i.e.,

$\begin{matrix}{{\langle{{E_{i\; n}(t)}{\sum\limits_{n = 1}^{\infty}{r^{n}e^{{- i}\; \beta \; n\; \sin \; {\omega_{m}{({t + \tau})}}}{E_{i\; n}^{*}\left( {t + \tau - {nT}} \right)}}}}\rangle} = {\sum\limits_{n = 1}^{\infty}{r^{n}{\langle e^{{- i}\; \beta \; n\; \sin \; {\omega_{m}{({t + \tau})}}}\rangle}\; {{\langle{{E_{i\; n}(t)}{E_{i\; n}^{*}\left( {t + \tau - {nT}} \right)}}\rangle}.}}}} & (37)\end{matrix}$

A Jacobi-Anger expansion can be applied to terms similar to the leftmostexpectation above, resulting in

$\begin{matrix}\begin{matrix}{{\langle e^{{\pm i}\; \beta \; n\; \sin \; {\omega_{m}{({t + \tau})}}}\rangle} = {\langle{\sum\limits_{p = {- \infty}}^{\infty}{{J_{p}\left( {\beta \; n} \right)}e^{{\pm {ip}}\; {\omega_{m}{({t + \tau})}}}}}\rangle}} \\{= {{J_{0}\left( {\beta \; n} \right)}.}}\end{matrix} & (38)\end{matrix}$

With some algebra, the following expressions can be obtained forS_(o,1)(ω), and S_(o,2) (ω), as a function of the input PSD S_(in)(ω):

$\begin{matrix}{{S_{o,1}(\omega)} = {\left\lbrack {\sum\limits_{n = 1}^{\infty}{r^{n}{J_{0}\left( {\beta \; n} \right)}e^{{- i}\; \omega \; {nT}}}} \right\rbrack \; {S_{i\; n}(\omega)}}} & (39) \\{{S_{o,2}(\omega)} = {\left\lbrack {\sum\limits_{n = 1}^{\infty}{r^{n}{J_{0}\left( {\beta \; n} \right)}e^{i\; \omega \; {nT}}}} \right\rbrack \; {S_{i\; n}(\omega)}}} & (40)\end{matrix}$

To calculate S_(o,3)(ω), uncorrelated terms are separated, similar to(37), resulting in the following simplification:

$\begin{matrix}{{\langle{e^{i\; \beta \; n\; \sin \; \omega_{m}t}e^{{- i}\; \beta \; m\; \sin \; {\omega_{m}{({t + \tau})}}}}\rangle} = {{\langle{\sum\limits_{p,{q = {- \infty}}}^{\infty}{{J_{p}\left( {\beta \; n} \right)}{J_{q}\left( {\beta \; m} \right)}e^{{i{({p - q})}}\omega_{m}t}e^{{- {iq}}\; \omega_{m}\tau}}}\rangle} = {\sum\limits_{p = {- \infty}}^{\infty}{{J_{p}\left( {\beta \; n} \right)}{J_{p}\left( {\beta \; m} \right)}{e^{{- i}\; p\; \omega_{m}\tau}.}}}}} & (41)\end{matrix}$

With some additional algebra, S_(o,3) (ω) can be expressed as a functionof S_(in)(ω)),

$\begin{matrix}{{S_{o,3}(\omega)} = {\sum\limits_{p = {- \infty}}^{\infty}{{S_{i\; n}\left( {\omega - {p\; \omega_{m}}} \right)} \times {\left\lbrack {\sum\limits_{n,{m = 1}}^{\infty}{r^{n + m}{J_{p}\left( {\beta \; n} \right)}{J_{p}\left( {\beta \; m} \right)}e^{i\; {\omega {({n - m})}}T}}} \right\rbrack.}}}} & (42)\end{matrix}$

Finally, the output PSD is

$\begin{matrix}{{{S_{out}(\omega)} = {{{\left( {1 - \gamma} \right)\left\lbrack {\left( {1 - k} \right) - {2k\; {Re}\left\{ {\chi_{0}(\omega)} \right\}}} \right\rbrack}{S_{i\; n}(\omega)}} + {\left( {1 - \gamma} \right)\frac{k^{2}}{1 - k}{\sum\limits_{p = {- \infty}}^{\infty}{{{\chi_{p}(\omega)}}^{2}{S_{i\; n}\left( {\omega - {p\; \omega_{m}}} \right)}}}}}},} & (43)\end{matrix}$

where the linewidth correction term χ_(p)(ω) is defined in (9).

Modulation Frequency Offset

The power in the pth comb line in the presence of modulation frequencyoffsets is derived, as defined above. First, in the presence ofmodulation frequency offsets, the cascaded modulation function, (2), canbe adjusted to include the modulation frequency offset ϕ_(m)=Δω_(m)T bynoting

$\begin{matrix}{{{\beta \; {F_{n}\left( {\omega_{m}t} \right)}} = {{\beta {\sum\limits_{i = 1}^{n}{\sin \; {\omega_{m}\left( {t - {iT}} \right)}}}} = {{{\beta \; {\sin \left( {\omega_{m}t} \right)}\left( {{\frac{1}{2}{\cot \left( {\varphi_{m}/2} \right)}} - \frac{\cos \left( {\left( {n + \frac{1}{2}} \right)\varphi_{m}} \right)}{2\; {\sin \left( {\varphi_{m}/2} \right)}}} \right)} - {\beta \; {\cos \left( {\omega_{m}t} \right)}\left( {{- \frac{1}{2}} + \frac{\sin \left( {\left( {n + \frac{1}{2}} \right)\varphi_{m}} \right)}{2\; {\sin \left( {\varphi_{m}/2} \right)}}} \right)}} = {{{\beta_{o}\left( {\varphi_{m},n} \right)}{\sin \left( {\omega_{m}t} \right)}} - {{\beta_{e}\left( {\varphi_{m},n} \right)}{\cos \left( {\omega_{m}t} \right)}}}}}},} & (44)\end{matrix}$

where in the second line Lagrange's trigonometric identities are usedwhere the final expression is simplified using β₀(ϕ_(m), n) andβ_(e)(ϕ_(m), n) as defined above. This expression can then be insertedinto (1) to find an expression for the output optical field in a similarmanner to that of (5):

$\begin{matrix}{{E_{out}(t)} = {{{\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{\hat{E}}_{i\; n}e^{i\; \omega_{0}t}} - {k\sqrt{\frac{1 - \gamma}{1 - k}}{\hat{E}}_{i\; n}e^{i\; \omega_{0}t} \times {\sum\limits_{n = 1}^{\infty}\left( {r^{n}e^{{- i}\; {\beta_{o}{({\varphi_{m},n})}}{\sin {({\omega_{m}t})}}}e^{{- i}\; {\beta_{o}{({\varphi_{m},n})}}\; {\cos {({\omega_{m}t})}}}} \right)}}} = {\sqrt{\left( {1 - \gamma} \right)\left( {1 - k} \right)}{\hat{E}}_{i\; n}{{e^{i\; \omega_{0}t}\left\lbrack {1 - {\frac{k}{1 - k}{\sum\limits_{p = {- \infty}}^{\infty}{e^{{ip}\; \omega_{m}t} \times {\sum\limits_{n = 1}^{\infty}{\sum\limits_{q = {- \infty}}^{\infty}{r^{n}i^{q}{J_{p - q}\left( {\beta_{o}\left( {\varphi_{m},n} \right)} \right)}{J_{q}\left( {\beta_{e}\left( {\varphi_{m},n} \right)} \right)}}}}}}}} \right\rbrack}.}}}} & (45)\end{matrix}$

From this output field, composed of equidistant frequencies spaced atthe modulation frequency, the output power in the pth comb line can becalculated, given by (15). When the modulator frequency is tuned exactlyto the resonator FSR (ϕ_(m)=0), this result reduces to (6).

Various exemplary embodiments described herein use lithium niobate forresonators and waveguides. However, it will be appreciated that avariety of eletrco-optic materials may be used in place of lithiumniobate, such as lithium tantalate. In general, any materials with anelectro-optic coefficient of at least 2 pm/V is suitable.

Various exemplary embodiments described herein use ring resonatorsresonators. However, it will be appreciated that alternative resonatorconfigurations may be substituted for one or more of the ring resonatorsin various embodiments. For example, a racetrack resonator may be used.

Various exemplary embodiments described herein include a thermal oxidesubstrate, such as SiO₂. However, it will be appreciated that a varietyof alternative substrates are suitable, including SiO₂, quartz, andsapphire. In general, any substrate with a low refractive index issuitable as a substrate. In this context, a low refractive indexmaterial is a material having a refractive index of n≤2.25 at normaltemperature and pressure (20° C./293.15 K/68° F. and 1 atm/14.696psi/101.325 kPa).

The descriptions of the various embodiments of the present disclosurehave been presented for purposes of illustration, but are not intendedto be exhaustive or limited to the embodiments disclosed. Manymodifications and variations will be apparent to those of ordinary skillin the art without departing from the scope and spirit of the describedembodiments. The terminology used herein was chosen to best explain theprinciples of the embodiments, the practical application or technicalimprovement over technologies found in the marketplace, or to enableothers of ordinary skill in the art to understand the embodimentsdisclosed herein.

1. A device comprising: a thermal oxide substrate; a microring resonatorcomprising lithium niobate, the microring resonator disposed on thethermal oxide substrate; a pump laser optically coupled to the microringresonator, wherein the microring resonator is adapted to emit a Kerrfrequency comb when receiving a pump mode from the pump laser. 2.(canceled)
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 33. A device comprising: athermal oxide substrate; a microring resonator comprising lithiumniobate, the microring resonator disposed on the thermal oxide substrateand having inner and outer edges; electrodes positioned along the innerand outer edges of the microring resonator, adapted to modulate therefractive index of the microring; a pump laser optically coupled to themicroring resonator, wherein the microring resonator is adapted to emitan electro-optical frequency comb when receiving a pump mode from thepump laser and when the electrodes are driven at a frequency equal to afree-spectral-range of the microring resonator.
 34. The device of claim33, wherein the pump laser is optically coupled to the microringresonator via a coupling ring resonator, the coupling ring resonatorhaving a free spectral range that is a non-integer multiple of a freespectral range of the microring resonator.
 35. The device of claim 34,further comprising: an output waveguide optically coupled to themicroring resonator.
 36. The device of claim 34, wherein the couplingring resonator has a free spectral range greater than that of themicroring resonator.
 37. The device of claim 33, wherein the microringresonator is further adapted to emit a Kerr frequency comb whenreceiving the pump mode from the pump laser.
 38. The device of claim 33,wherein the electro-optical frequency comb spans at least 10 nm.
 39. Thedevice of claim 33, wherein the electro-optical frequency comb hasspacing of 1 GHz to 300 GHz.
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 41. The device of claim 33,wherein the electrodes comprise gold or copper.
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 43. Thedevice of claim 33, wherein the microring resonator has a Q factor of atleast 500,000.
 44. The device of claim 33, wherein the electrodes arepositioned at least 1.5 μm from the edges of the microring resonator.45. (canceled)
 46. The device of claim 33, wherein the thermal oxidesubstrate has a thickness of about 1 μm.
 47. The device of claim 33,wherein the thermal oxide substrate has a thickness of about 2 μm. 48.The device of claim 33, wherein the electrodes are driven at a frequencyof about 10 GHz.
 49. The device of claim 33, wherein the electrodes aredriven at a power of about 10 mW.
 50. The device of claim 33, whereinthe pump laser has a power of 0.1 mW to 3 W.
 51. The device of claim 33,wherein the pump laser has a power of from 2 mW to 100 mW. 52.(canceled)
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 55. The device of claim 33,wherein the electro-optical frequency comb has a center wavelength of380 nm to 5000 nm.
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 58. The device of claim33, wherein the microring resonator comprises a ridge portion extendingfrom a slab portion, the ridge portion having a height perpendicular tothe slab portion and a width parallel to the slab portion.
 59. Thedevice of claim 58, wherein the slab portion has a thickness of 5 nm to1000 nm.
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 61. The device of claim 58, wherein the ridgeportion has a height of about 350 nm.
 62. The device of claim 58,wherein the ridge portion has a width of 100 nm to 5000 nm. 63.(canceled)
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 65. The device of claim 58, wherein the ridgeportion has a cross sectional area less than 5 μm².
 66. (canceled) 67.The device of claim 33, wherein the microring resonator is air-clad. 68.The device of claim 33, wherein the microring resonator is clad withSiO₂.
 69. The device of claim 33, further comprising: an inductorelectrically coupled to the electrodes.
 70. The device of claim 69,wherein the inductor is adapted to form a microwave resonator having aresonant frequency, the resonant frequency being an integer multiple ofa free-spectral range of the microring resonator.
 71. A devicecomprising: a substrate; a microring resonator comprising anelectro-optic material, the microring resonator disposed on thesubstrate; a pump laser optically coupled to the microring resonator,wherein the microring resonator is adapted to emit a Kerr frequency combwhen receiving a pump mode from the pump laser.
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 109. A device comprising: a substrate; a resonator comprisingan electro-optic material, the resonator disposed on the substrate;electrodes positioned along the resonator with at least a portion of theresonator disposed between the electrodes, the electrodes adapted tomodulate the refractive index of the resonator; a pump laser opticallycoupled to the resonator, wherein the resonator is adapted to emit anelectro-optical frequency comb when receiving a pump mode from the pumplaser and when the electrodes are driven at a frequency, the frequencybeing an integer multiple of a free-spectral-range of the resonator.110. The device of claim 109, wherein the substrate comprises a thermaloxide.
 111. The device of claim 109, wherein the substrate comprisesSiO₂, quartz, or sapphire.
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 114. Thedevice of claim 109, wherein the electro-optic material compriseslithium niobate or lithium tantalate.
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 116. The device ofclaim 109, wherein the electro-optic material has an electro-opticcoefficient of at least 2 pm/V.
 117. The device of claim 109, whereinthe resonator comprises a racetrack resonator.
 118. The device of claim117, wherein the racetrack resonator has a minor axis measuring 20 μm to2000 μm and a perpendicular major axis measuring 0.1 mm to 20 mm. 119.(canceled)
 120. The device of claim 119, wherein the major axis isperpendicular to an extraordinary axis of the electro-optic material.121. The device of claim 109, wherein the resonator comprises a ringresonator.
 122. The device of claim 109, wherein the resonator comprisesa ring resonator or a racetrack resonator, the resonator has inner andouter edges, a first surface in contact with the substrate, and a secondsurface parallel to the first surface, the electrodes are positionedalong the first and second surfaces of the resonator.
 123. The device ofclaim 109, wherein the resonator comprises a ring resonator or aracetrack resonator, the resonator has inner and outer edges, a firstsurface in contact with the substrate, and a second surface parallel tothe first surface, a first electrode is positioned along the outer edgeof the resonator, a second electrode is positioned along the secondsurface of the resonator.
 124. The device of claim 109, wherein the pumplaser is optically coupled to the resonator via a coupling resonator,the coupling resonator having a free spectral range that is anon-integer multiple of a free spectral range of the resonator.
 125. Thedevice of claim 124, wherein the coupling resonator comprises a ringresonator.
 126. The device of claim 124, further comprising: an outputwaveguide optically coupled to the resonator.
 127. The device of claim124, wherein the coupling resonator has a free spectral range greaterthan that of the resonator.
 128. The device of claim 109, wherein theresonator is further adapted to emit a Kerr frequency comb whenreceiving the pump mode from the pump laser.
 129. The device of claim109, wherein the electro-optical frequency comb spans at least 10 nm.130. The device of claim 109, wherein the electro-optical frequency combhas spacing of 1 GHz to 300 GHz.
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 132. The device ofclaim 109, wherein the electrodes comprise gold or copper. 133.(canceled)
 134. The device of claim 109, wherein the resonator has a Qfactor of at least 500,000.
 135. The device of claim 109, wherein theelectrodes are positioned at least 1.5 μm from the edges of theresonator.
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 137. The device of claim 109, wherein thesubstrate has a thickness of about 1 μm.
 138. The device of claim 109,wherein the substrate has a thickness of about 2 μm.
 139. The device ofclaim 109, wherein the electrodes are driven at a frequency of about 10GHz.
 140. The device of claim 109, wherein the electrodes are driven ata power of about 10 mW.
 141. The device of claim 109, wherein the pumplaser has a power of 0.1 mW to 3 W.
 142. The device of claim 109,wherein the pump laser has a power of from 2 mW to 100 mW. 143.(canceled)
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 146. The device of claim 109,wherein the electro-optical frequency comb has a center wavelength of380 nm to 5000 nm.
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 149. The device ofclaim 109, wherein the resonator comprises a ridge portion extendingfrom a slab portion, the ridge portion having a height perpendicular tothe slab portion and a width parallel to the slab portion.
 150. Thedevice of claim 149, wherein the slab portion has a thickness of 5 nm to1000 nm.
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 152. The device of claim 149, wherein the ridgeportion has a height of about 350 nm.
 153. The device of claim 149,wherein the ridge portion has a width of 100 nm to 5000 nm. 154.(canceled)
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 156. The device of claim 149, wherein theridge portion has a cross sectional area less than 5 μm². 157.(canceled)
 158. The device of claim 109, wherein the resonator isair-clad.
 159. The device of claim 109, wherein the resonator is cladwith SiO₂.
 160. The device of claim 109, further comprising: an inductorelectrically coupled to the electrodes.
 161. The device of claim 160,wherein the inductor is adapted to form a microwave resonator having aresonant frequency, the resonant frequency being an integer multiple ofa free-spectral range of the microring resonator.
 162. A method ofgenerating a Kerr frequency comb, the method comprising: receiving apump mode from a pump laser by a microring resonator, wherein themicroring resonator comprising an electro-optic material, the microringresonator disposed on a substrate, the pump laser optically coupled tothe microring resonator, the microring resonator is adapted to emit aKerr frequency comb when receiving a pump mode from the pump laser. 163.(canceled)
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 200. A method of generating an electro-optical frequencycomb, the method comprising: receiving a pump mode from a pump laser bya resonator, wherein the resonator comprises an electro-optic material,the resonator is disposed on a substrate; driving electrodes at afrequency, the frequency being an integer multiple of afree-spectral-range of the resonator, wherein the electrodes arepositioned along the resonator with at least a portion of the resonatordisposed between the electrodes, the electrodes adapted to modulate therefractive index of the resonator.
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